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Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 6n}{n + 3}$ converge?
1. No: the terms tend to 6 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 6
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 5n}{n + 8}$ converge?
1. No: the terms tend to 5 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 5
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 5n}{n + 2}$ converge?
1. No: the terms tend to 5 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 5
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 2n}{n + 1}$ converge?
1. No: the terms tend to 2 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 2
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 8n}{n + 4}$ converge?
1. No: the terms tend to 8 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 8
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 3n}{n + 6}$ converge?
1. No: the terms tend to 3 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 3
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 4n}{n + 6}$ converge?
1. No: the terms tend to 4 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 4
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 7n}{n + 8}$ converge?
1. No: the terms tend to 7 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 7
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 6n}{n + 7}$ converge?
1. No: the terms tend to 6 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 6
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 6n}{n + 4}$ converge?
1. No: the terms tend to 6 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 6
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 3n}{n + 7}$ converge?
1. No: the terms tend to 3 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 3
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 5n}{n + 7}$ converge?
1. No: the terms tend to 5 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 5
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 1n}{n + 3}$ converge?
1. No: the terms tend to 1 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 1
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 7n}{n + 6}$ converge?
1. No: the terms tend to 7 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 7
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 5n}{n + 4}$ converge?
1. No: the terms tend to 5 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 5
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 3n}{n + 9}$ converge?
1. No: the terms tend to 3 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 3
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 3n}{n + 8}$ converge?
1. No: the terms tend to 3 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 3
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 9n}{n + 2}$ converge?
1. No: the terms tend to 9 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 9
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 9n}{n + 1}$ converge?
1. No: the terms tend to 9 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 9
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 7n}{n + 5}$ converge?
1. No: the terms tend to 7 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 7
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 8n}{n + 5}$ converge?
1. No: the terms tend to 8 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 8
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 4n}{n + 9}$ converge?
1. No: the terms tend to 4 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 4
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 7n}{n + 7}$ converge?
1. No: the terms tend to 7 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 7
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 2n}{n + 6}$ converge?
1. No: the terms tend to 2 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 2
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 9n}{n + 4}$ converge?
1. No: the terms tend to 9 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 9
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 9n}{n + 6}$ converge?
1. No: the terms tend to 9 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 9
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 7n}{n + 2}$ converge?
1. No: the terms tend to 7 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 7
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 2n}{n + 4}$ converge?
1. No: the terms tend to 2 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 2
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 4n}{n + 1}$ converge?
1. No: the terms tend to 4 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 4
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 2n}{n + 2}$ converge?
1. No: the terms tend to 2 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 2
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 1n}{n + 5}$ converge?
1. No: the terms tend to 1 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 1
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 4n}{n + 7}$ converge?
1. No: the terms tend to 4 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 4
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 3n}{n + 3}$ converge?
1. No: the terms tend to 3 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 3
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 1n}{n + 8}$ converge?
1. No: the terms tend to 1 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 1
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 3n}{n + 4}$ converge?
1. No: the terms tend to 3 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 3
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 6n}{n + 5}$ converge?
1. No: the terms tend to 6 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 6
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 3n}{n + 1}$ converge?
1. No: the terms tend to 3 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 3
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 5n}{n + 5}$ converge?
1. No: the terms tend to 5 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 5
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 2n}{n + 5}$ converge?
1. No: the terms tend to 2 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 2
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 2n}{n + 7}$ converge?
1. No: the terms tend to 2 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 2
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 4n}{n + 2}$ converge?
1. No: the terms tend to 4 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 4
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 8n}{n + 2}$ converge?
1. No: the terms tend to 8 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 8
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 8n}{n + 3}$ converge?
1. No: the terms tend to 8 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 8
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 6n}{n + 8}$ converge?
1. No: the terms tend to 6 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 6
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 6n}{n + 1}$ converge?
1. No: the terms tend to 6 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 6
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 1n}{n + 6}$ converge?
1. No: the terms tend to 1 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 1
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 8n}{n + 7}$ converge?
1. No: the terms tend to 8 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 8
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 5n}{n + 9}$ converge?
1. No: the terms tend to 5 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 5
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 4n}{n + 3}$ converge?
1. No: the terms tend to 4 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 4
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 8n}{n + 8}$ converge?
1. No: the terms tend to 8 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 8
3. The nth-term test cannot decide
$\displaystyle\sum_{n=1}^{\infty} \dfrac{ 4 }{n}$ : its terms tend to 0. What does that tell you?
1. Nothing by itself: this series in fact diverges (harmonic series)
2. The series converges
3. The sum is 0
$\displaystyle\sum_{n=1}^{\infty} \dfrac{ 5 }{n}$ : its terms tend to 0. What does that tell you?
1. Nothing by itself: this series in fact diverges (harmonic series)
2. The series converges
3. The sum is 0
$\displaystyle\sum_{n=1}^{\infty} \dfrac{ 3 }{n}$ : its terms tend to 0. What does that tell you?
1. Nothing by itself: this series in fact diverges (harmonic series)
2. The series converges
3. The sum is 0
$\displaystyle\sum_{n=1}^{\infty} \dfrac{ 8 }{n}$ : its terms tend to 0. What does that tell you?
1. Nothing by itself: this series in fact diverges (harmonic series)
2. The series converges
3. The sum is 0
$\displaystyle\sum_{n=1}^{\infty} \dfrac{ 6 }{n}$ : its terms tend to 0. What does that tell you?
1. Nothing by itself: this series in fact diverges (harmonic series)
2. The series converges
3. The sum is 0
$\displaystyle\sum_{n=1}^{\infty} \dfrac{ 9 }{n}$ : its terms tend to 0. What does that tell you?
1. Nothing by itself: this series in fact diverges (harmonic series)
2. The series converges
3. The sum is 0
$\displaystyle\sum_{n=1}^{\infty} \dfrac{ 7 }{n}$ : its terms tend to 0. What does that tell you?
1. Nothing by itself: this series in fact diverges (harmonic series)
2. The series converges
3. The sum is 0
$\displaystyle\sum_{n=1}^{\infty} \dfrac{ 2 }{n}$ : its terms tend to 0. What does that tell you?
1. Nothing by itself: this series in fact diverges (harmonic series)
2. The series converges
3. The sum is 0
$\displaystyle\sum_{n=1}^{\infty} \dfrac{ 1 }{n}$ : its terms tend to 0. What does that tell you?
1. Nothing by itself: this series in fact diverges (harmonic series)
2. The series converges
3. The sum is 0
Find $\displaystyle\sum_{n=1}^{\infty} \left(\dfrac{ 9 }{n} - \dfrac{ 9 }{n + 1}\right)$ .
Use the integral test on $\displaystyle\sum_{n=1}^{\infty} 7n e^{-n^2}$ .
1. It converges: the integral of 7xe^(−x²) from 1 to ∞ is finite
2. It diverges, because n grows
3. The integral test does not apply
Use the integral test on $\displaystyle\sum_{n=1}^{\infty} 4n e^{-n^2}$ .
1. It converges: the integral of 4xe^(−x²) from 1 to ∞ is finite
2. It diverges, because n grows
3. The integral test does not apply
Use the integral test on $\displaystyle\sum_{n=1}^{\infty} 9n e^{-n^2}$ .
1. It converges: the integral of 9xe^(−x²) from 1 to ∞ is finite
2. It diverges, because n grows
3. The integral test does not apply
Use the integral test on $\displaystyle\sum_{n=1}^{\infty} 3n e^{-n^2}$ .
1. It converges: the integral of 3xe^(−x²) from 1 to ∞ is finite
2. It diverges, because n grows
3. The integral test does not apply
Use the integral test on $\displaystyle\sum_{n=1}^{\infty} 6n e^{-n^2}$ .
1. It converges: the integral of 6xe^(−x²) from 1 to ∞ is finite
2. It diverges, because n grows
3. The integral test does not apply
Use the integral test on $\displaystyle\sum_{n=1}^{\infty} 2n e^{-n^2}$ .
1. It converges: the integral of 2xe^(−x²) from 1 to ∞ is finite
2. It diverges, because n grows
3. The integral test does not apply
Use the integral test on $\displaystyle\sum_{n=1}^{\infty} 8n e^{-n^2}$ .
1. It converges: the integral of 8xe^(−x²) from 1 to ∞ is finite
2. It diverges, because n grows
3. The integral test does not apply
Use the integral test on $\displaystyle\sum_{n=1}^{\infty} 5n e^{-n^2}$ .
1. It converges: the integral of 5xe^(−x²) from 1 to ∞ is finite
2. It diverges, because n grows
3. The integral test does not apply
Use the integral test on $\displaystyle\sum_{n=1}^{\infty} 1n e^{-n^2}$ .
1. It converges: the integral of 1xe^(−x²) from 1 to ∞ is finite
2. It diverges, because n grows
3. The integral test does not apply
The integral test bounds $\displaystyle\sum_{k=2}^{\infty} \dfrac{1}{k^{ 6 } }$ by $\displaystyle\int_1^{\infty} \dfrac{dx}{x^{ 6 } }$ . Find that integral.
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^2 + 7}$ converge?
1. Yes: it is smaller than the convergent ∑ 1/n²
2. No: it is smaller than the divergent ∑ 1/n
3. No test applies
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^2 + 8}$ converge?
1. Yes: it is smaller than the convergent ∑ 1/n²
2. No: it is smaller than the divergent ∑ 1/n
3. No test applies
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^2 + 9}$ converge?
1. Yes: it is smaller than the convergent ∑ 1/n²
2. No: it is smaller than the divergent ∑ 1/n
3. No test applies
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^2 + 2}$ converge?
1. Yes: it is smaller than the convergent ∑ 1/n²
2. No: it is smaller than the divergent ∑ 1/n
3. No test applies
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^2 + 5}$ converge?
1. Yes: it is smaller than the convergent ∑ 1/n²
2. No: it is smaller than the divergent ∑ 1/n
3. No test applies
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^2 + 6}$ converge?
1. Yes: it is smaller than the convergent ∑ 1/n²
2. No: it is smaller than the divergent ∑ 1/n
3. No test applies
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^2 + 1}$ converge?
1. Yes: it is smaller than the convergent ∑ 1/n²
2. No: it is smaller than the divergent ∑ 1/n
3. No test applies
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^2 + 4}$ converge?
1. Yes: it is smaller than the convergent ∑ 1/n²
2. No: it is smaller than the divergent ∑ 1/n
3. No test applies
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^2 + 3}$ converge?
1. Yes: it is smaller than the convergent ∑ 1/n²
2. No: it is smaller than the divergent ∑ 1/n
3. No test applies
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{\sqrt{n} + 8}$ converge?
1. No: it behaves like ∑ 1/√n, which diverges (limit comparison)
2. Yes: it is smaller than ∑ 1/√n
3. Yes: its terms tend to 0
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{\sqrt{n} + 6}$ converge?
1. No: it behaves like ∑ 1/√n, which diverges (limit comparison)
2. Yes: it is smaller than ∑ 1/√n
3. Yes: its terms tend to 0
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{\sqrt{n} + 1}$ converge?
1. No: it behaves like ∑ 1/√n, which diverges (limit comparison)
2. Yes: it is smaller than ∑ 1/√n
3. Yes: its terms tend to 0
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{\sqrt{n} + 4}$ converge?
1. No: it behaves like ∑ 1/√n, which diverges (limit comparison)
2. Yes: it is smaller than ∑ 1/√n
3. Yes: its terms tend to 0
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{\sqrt{n} + 7}$ converge?
1. No: it behaves like ∑ 1/√n, which diverges (limit comparison)
2. Yes: it is smaller than ∑ 1/√n
3. Yes: its terms tend to 0
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{\sqrt{n} + 2}$ converge?
1. No: it behaves like ∑ 1/√n, which diverges (limit comparison)
2. Yes: it is smaller than ∑ 1/√n
3. Yes: its terms tend to 0
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{\sqrt{n} + 3}$ converge?
1. No: it behaves like ∑ 1/√n, which diverges (limit comparison)
2. Yes: it is smaller than ∑ 1/√n
3. Yes: its terms tend to 0
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{\sqrt{n} + 5}$ converge?
1. No: it behaves like ∑ 1/√n, which diverges (limit comparison)
2. Yes: it is smaller than ∑ 1/√n
3. Yes: its terms tend to 0
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{\sqrt{n} + 9}$ converge?
1. No: it behaves like ∑ 1/√n, which diverges (limit comparison)
2. Yes: it is smaller than ∑ 1/√n
3. Yes: its terms tend to 0
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{2n + 1}{n^3 + 8}$ converge?
1. Yes: it behaves like ∑ 2/n², a convergent p-series
2. No: the top grows
3. No: it behaves like ∑ 1/n
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{2n + 1}{n^3 + 4}$ converge?
1. Yes: it behaves like ∑ 2/n², a convergent p-series
2. No: the top grows
3. No: it behaves like ∑ 1/n
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{2n + 1}{n^3 + 2}$ converge?
1. Yes: it behaves like ∑ 2/n², a convergent p-series
2. No: the top grows
3. No: it behaves like ∑ 1/n
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{2n + 1}{n^3 + 3}$ converge?
1. Yes: it behaves like ∑ 2/n², a convergent p-series
2. No: the top grows
3. No: it behaves like ∑ 1/n
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{2n + 1}{n^3 + 1}$ converge?
1. Yes: it behaves like ∑ 2/n², a convergent p-series
2. No: the top grows
3. No: it behaves like ∑ 1/n
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{2n + 1}{n^3 + 6}$ converge?
1. Yes: it behaves like ∑ 2/n², a convergent p-series
2. No: the top grows
3. No: it behaves like ∑ 1/n
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{2n + 1}{n^3 + 5}$ converge?
1. Yes: it behaves like ∑ 2/n², a convergent p-series
2. No: the top grows
3. No: it behaves like ∑ 1/n
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{2n + 1}{n^3 + 7}$ converge?
1. Yes: it behaves like ∑ 2/n², a convergent p-series
2. No: the top grows
3. No: it behaves like ∑ 1/n
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{2n + 1}{n^3 + 9}$ converge?
1. Yes: it behaves like ∑ 2/n², a convergent p-series
2. No: the top grows
3. No: it behaves like ∑ 1/n
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1} 7 }{n}$ converge?
1. Yes, conditionally: the terms alternate, shrink and tend to 0
2. No: ∑ 7/n diverges
3. Yes, absolutely
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1} 6 }{n}$ converge?
1. Yes, conditionally: the terms alternate, shrink and tend to 0
2. No: ∑ 6/n diverges
3. Yes, absolutely
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1} 8 }{n}$ converge?
1. Yes, conditionally: the terms alternate, shrink and tend to 0
2. No: ∑ 8/n diverges
3. Yes, absolutely
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1} 9 }{n}$ converge?
1. Yes, conditionally: the terms alternate, shrink and tend to 0
2. No: ∑ 9/n diverges
3. Yes, absolutely
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1} 2 }{n}$ converge?
1. Yes, conditionally: the terms alternate, shrink and tend to 0
2. No: ∑ 2/n diverges
3. Yes, absolutely
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1} 1 }{n}$ converge?
1. Yes, conditionally: the terms alternate, shrink and tend to 0
2. No: ∑ 1/n diverges
3. Yes, absolutely
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1} 4 }{n}$ converge?
1. Yes, conditionally: the terms alternate, shrink and tend to 0
2. No: ∑ 4/n diverges
3. Yes, absolutely
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1} 5 }{n}$ converge?
1. Yes, conditionally: the terms alternate, shrink and tend to 0
2. No: ∑ 5/n diverges
3. Yes, absolutely
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1} 3 }{n}$ converge?
1. Yes, conditionally: the terms alternate, shrink and tend to 0
2. No: ∑ 3/n diverges
3. Yes, absolutely
$S = \displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1} }{n^2}$ is approximated by its first 8 terms. Give the alternating-series bound on the error.
Find the partial sum $S_3$ of $\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1} 1 }{n}$ .
For $a_n = \dfrac{n}{ 8^n}$ , find $L = \displaystyle\lim_{n \to \infty} \left|\dfrac{a_{n+1} }{a_n}\right|$ .
For $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^{ 2 } }$ , the ratio test gives $L = 1$ . What does it tell you?
1. Nothing: the ratio test is inconclusive when L = 1
2. The series converges
3. The series diverges
For $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^{ 8 } }$ , the ratio test gives $L = 1$ . What does it tell you?
1. Nothing: the ratio test is inconclusive when L = 1
2. The series converges
3. The series diverges
For $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^{ 1 } }$ , the ratio test gives $L = 1$ . What does it tell you?
1. Nothing: the ratio test is inconclusive when L = 1
2. The series converges
3. The series diverges
For $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^{ 6 } }$ , the ratio test gives $L = 1$ . What does it tell you?
1. Nothing: the ratio test is inconclusive when L = 1
2. The series converges
3. The series diverges
For $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^{ 3 } }$ , the ratio test gives $L = 1$ . What does it tell you?
1. Nothing: the ratio test is inconclusive when L = 1
2. The series converges
3. The series diverges
For $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^{ 5 } }$ , the ratio test gives $L = 1$ . What does it tell you?
1. Nothing: the ratio test is inconclusive when L = 1
2. The series converges
3. The series diverges
For $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^{ 4 } }$ , the ratio test gives $L = 1$ . What does it tell you?
1. Nothing: the ratio test is inconclusive when L = 1
2. The series converges
3. The series diverges
For $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^{ 9 } }$ , the ratio test gives $L = 1$ . What does it tell you?
1. Nothing: the ratio test is inconclusive when L = 1
2. The series converges
3. The series diverges
For $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^{ 7 } }$ , the ratio test gives $L = 1$ . What does it tell you?
1. Nothing: the ratio test is inconclusive when L = 1
2. The series converges
3. The series diverges
How does $\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^n 1 }{n^2}$ converge?
1. Absolutely: ∑ 1/n² converges
2. Only conditionally
3. It diverges
How does $\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^n 5 }{n^2}$ converge?
1. Absolutely: ∑ 5/n² converges
2. Only conditionally
3. It diverges
How does $\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^n 6 }{n^2}$ converge?
1. Absolutely: ∑ 6/n² converges
2. Only conditionally
3. It diverges
How does $\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^n 4 }{n^2}$ converge?
1. Absolutely: ∑ 4/n² converges
2. Only conditionally
3. It diverges
How does $\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^n 3 }{n^2}$ converge?
1. Absolutely: ∑ 3/n² converges
2. Only conditionally
3. It diverges
How does $\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^n 9 }{n^2}$ converge?
1. Absolutely: ∑ 9/n² converges
2. Only conditionally
3. It diverges
How does $\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^n 7 }{n^2}$ converge?
1. Absolutely: ∑ 7/n² converges
2. Only conditionally
3. It diverges
How does $\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^n 8 }{n^2}$ converge?
1. Absolutely: ∑ 8/n² converges
2. Only conditionally
3. It diverges
How does $\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^n 2 }{n^2}$ converge?
1. Absolutely: ∑ 2/n² converges
2. Only conditionally
3. It diverges
Which test suits $\displaystyle\sum_{n=1}^{\infty} \dfrac{n!}{ 9^n}$ best?
1. The ratio test
2. Comparison with a p-series
3. The alternating series test
Which test suits $\displaystyle\sum_{n=1}^{\infty} \dfrac{n!}{ 8^n}$ best?
1. The ratio test
2. Comparison with a p-series
3. The alternating series test
Which test suits $\displaystyle\sum_{n=1}^{\infty} \dfrac{n!}{ 4^n}$ best?
1. The ratio test
2. Comparison with a p-series
3. The alternating series test
Which test suits $\displaystyle\sum_{n=1}^{\infty} \dfrac{n!}{ 2^n}$ best?
1. The ratio test
2. Comparison with a p-series
3. The alternating series test
Which test suits $\displaystyle\sum_{n=1}^{\infty} \dfrac{n!}{ 5^n}$ best?
1. The ratio test
2. Comparison with a p-series
3. The alternating series test
Which test suits $\displaystyle\sum_{n=1}^{\infty} \dfrac{n!}{ 7^n}$ best?
1. The ratio test
2. Comparison with a p-series
3. The alternating series test
Which test suits $\displaystyle\sum_{n=1}^{\infty} \dfrac{n!}{ 6^n}$ best?
1. The ratio test
2. Comparison with a p-series
3. The alternating series test
Which test suits $\displaystyle\sum_{n=1}^{\infty} \dfrac{n!}{ 3^n}$ best?
1. The ratio test
2. Comparison with a p-series
3. The alternating series test
Which test shows that $\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^n 5 }{\sqrt{n} }$ converges?
1. The alternating series test
2. The p-test with p = ½
3. The ratio test
Which test shows that $\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^n 6 }{\sqrt{n} }$ converges?
1. The alternating series test
2. The p-test with p = ½
3. The ratio test
Which test shows that $\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^n 9 }{\sqrt{n} }$ converges?
1. The alternating series test
2. The p-test with p = ½
3. The ratio test
Which test shows that $\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^n 1 }{\sqrt{n} }$ converges?
1. The alternating series test
2. The p-test with p = ½
3. The ratio test
Which test shows that $\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^n 3 }{\sqrt{n} }$ converges?
1. The alternating series test
2. The p-test with p = ½
3. The ratio test
Which test shows that $\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^n 8 }{\sqrt{n} }$ converges?
1. The alternating series test
2. The p-test with p = ½
3. The ratio test
Which test shows that $\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^n 2 }{\sqrt{n} }$ converges?
1. The alternating series test
2. The p-test with p = ½
3. The ratio test
Which test shows that $\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^n 4 }{\sqrt{n} }$ converges?
1. The alternating series test
2. The p-test with p = ½
3. The ratio test
Which test shows that $\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^n 7 }{\sqrt{n} }$ converges?
1. The alternating series test
2. The p-test with p = ½
3. The ratio test
Which test settles $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^{ 6 } }$ fastest?
1. The p-series test: p = 6 > 1, so it converges
2. The nth-term test
3. The ratio test
Which test settles $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^{ 4 } }$ fastest?
1. The p-series test: p = 4 > 1, so it converges
2. The nth-term test
3. The ratio test
Which test settles $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^{ 8 } }$ fastest?
1. The p-series test: p = 8 > 1, so it converges
2. The nth-term test
3. The ratio test
Which test settles $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^{ 7 } }$ fastest?
1. The p-series test: p = 7 > 1, so it converges
2. The nth-term test
3. The ratio test
Which test settles $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^{ 2 } }$ fastest?
1. The p-series test: p = 2 > 1, so it converges
2. The nth-term test
3. The ratio test
Which test settles $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^{ 3 } }$ fastest?
1. The p-series test: p = 3 > 1, so it converges
2. The nth-term test
3. The ratio test
Which test settles $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^{ 5 } }$ fastest?
1. The p-series test: p = 5 > 1, so it converges
2. The nth-term test
3. The ratio test
Which test settles $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^{ 9 } }$ fastest?
1. The p-series test: p = 9 > 1, so it converges
2. The nth-term test
3. The ratio test

Answers

No: the terms tend to 6 ≠ 0, so it diverges (nth-term test)
Nothing by itself: this series in fact diverges (harmonic series)
$9$
It converges: the integral of 7xe^(−x²) from 1 to ∞ is finite
$\frac{1}{5}$
Yes: it is smaller than the convergent ∑ 1/n²
No: it behaves like ∑ 1/√n, which diverges (limit comparison)
Yes: it behaves like ∑ 2/n², a convergent p-series
Yes, conditionally: the terms alternate, shrink and tend to 0
$\frac{1}{81}$
$\frac{5}{6}$
$\frac{1}{8}$
Nothing: the ratio test is inconclusive when L = 1
Absolutely: ∑ 1/n² converges
The ratio test
The alternating series test
The p-series test: p = 6 > 1, so it converges