The integral test and p-series

About this lesson

Calculus BC (AP-aligned) › Infinite series and convergence tests

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The integral test and p-series

Warm-up

Three quick questions from earlier lessons.

Discover

Draw rectangles of width 1 and heights 1n2\tfrac{1}{n^2} from n=2n = 2 on. They fit under y=1x2y = \tfrac{1}{x^2} from 1 to ∞, whose area is 1. So is ∑n=2∞1n2\displaystyle\sum_{n=2}^{\infty} \tfrac{1}{n^2} finite?

  1. Yes, less than 1
  2. No, there are infinitely many terms

Key idea

A series and an integral share their fate.

Integral test: if ff is positive, continuous and decreasing with an=f(n)a_n = f(n), then ∑an\sum a_n and ∫1∞f(x) dx\displaystyle\int_1^{\infty} f(x)\,dx both converge or both diverge. p-series: ∑1np\displaystyle\sum \dfrac{1}{n^p} converges exactly when p>1p > 1.

Worked example

Does ∑1n1.1\displaystyle\sum \dfrac{1}{n^{1.1}} converge? And ∑1n0.9\displaystyle\sum \dfrac{1}{n^{0.9}}?

∑1n1.1\displaystyle\sum \dfrac{1}{n^{1.1}} converges
p = 1.1 > 1.
∑1n0.9\displaystyle\sum \dfrac{1}{n^{0.9}} diverges
p = 0.9 ≤ 1.

Your turn, guided

Your turn: the p of ∑1nn\displaystyle\sum \dfrac{1}{n\sqrt{n}}.

p=32p = \tfrac{3}{2}: converges
n · n^{1/2} = n^{3/2}.

Practice ★

  1. Which series converge? Tick all that do.
    1. ∑n=1∞1n2\displaystyle\sum_{n=1}^{\infty} \dfrac{ 1 }{n^2}
    2. ∑n=1∞1n3/2\displaystyle\sum_{n=1}^{\infty} \dfrac{ 1 }{n^{3/2} }
    3. ∑n=1∞1n\displaystyle\sum_{n=1}^{\infty} \dfrac{ 1 }{\sqrt{n} }
    4. ∑n=1∞1n\displaystyle\sum_{n=1}^{\infty} \dfrac{ 1 }{n}
  2. Which series converge? Tick all that do.
    1. ∑n=1∞4n2\displaystyle\sum_{n=1}^{\infty} \dfrac{ 4 }{n^2}
    2. ∑n=1∞4n3/2\displaystyle\sum_{n=1}^{\infty} \dfrac{ 4 }{n^{3/2} }
    3. ∑n=1∞4n\displaystyle\sum_{n=1}^{\infty} \dfrac{ 4 }{\sqrt{n} }
    4. ∑n=1∞4n\displaystyle\sum_{n=1}^{\infty} \dfrac{ 4 }{n}

Watch out!

Practice ★★

  1. Use the integral test on ∑n=1∞7ne−n2\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
  1. The integral test bounds ∑k=2∞1k6\displaystyle\sum_{k=2}^{\infty} \dfrac{1}{k^{ 6 } } by ∫1∞dxx6\displaystyle\int_1^{\infty} \dfrac{dx}{x^{ 6 } }. Find that integral.
  2. The integral test bounds ∑k=2∞1k2\displaystyle\sum_{k=2}^{\infty} \dfrac{1}{k^{ 2 } } by ∫1∞dxx2\displaystyle\int_1^{\infty} \dfrac{dx}{x^{ 2 } }. Find that integral.

Exit ticket

Three questions to check you have got it.

  1. Which series converge? Tick all that do.
    1. ∑n=1∞1n2\displaystyle\sum_{n=1}^{\infty} \dfrac{ 1 }{n^2}
    2. ∑n=1∞1n3/2\displaystyle\sum_{n=1}^{\infty} \dfrac{ 1 }{n^{3/2} }
    3. ∑n=1∞1n\displaystyle\sum_{n=1}^{\infty} \dfrac{ 1 }{\sqrt{n} }
    4. ∑n=1∞1n\displaystyle\sum_{n=1}^{\infty} \dfrac{ 1 }{n}
  2. Which series converge? Tick all that do.
    1. ∑n=1∞4n2\displaystyle\sum_{n=1}^{\infty} \dfrac{ 4 }{n^2}
    2. ∑n=1∞4n3/2\displaystyle\sum_{n=1}^{\infty} \dfrac{ 4 }{n^{3/2} }
    3. ∑n=1∞4n\displaystyle\sum_{n=1}^{\infty} \dfrac{ 4 }{\sqrt{n} }
    4. ∑n=1∞4n\displaystyle\sum_{n=1}^{\infty} \dfrac{ 4 }{n}
  3. Which series converge? Tick all that do.
    1. ∑n=1∞8n2\displaystyle\sum_{n=1}^{\infty} \dfrac{ 8 }{n^2}
    2. ∑n=1∞8n3/2\displaystyle\sum_{n=1}^{\infty} \dfrac{ 8 }{n^{3/2} }
    3. ∑n=1∞8n\displaystyle\sum_{n=1}^{\infty} \dfrac{ 8 }{\sqrt{n} }
    4. ∑n=1∞8n\displaystyle\sum_{n=1}^{\infty} \dfrac{ 8 }{n}

Summary card

Integral test, p-series

  • Positive decreasing terms: compare with ∫1∞f\int_1^{\infty} f.
  • ∑1/np\sum 1/n^p: converges iff p>1p > 1.

Watch out: p=1p = 1 diverges.

End of the lesson

Next lesson: Comparison tests Back to the chapter

Answers: The integral test and p-series

A

  1. ∑n=1∞1n2\displaystyle\sum_{n=1}^{\infty} \dfrac{ 1 }{n^2} · ∑n=1∞1n3/2\displaystyle\sum_{n=1}^{\infty} \dfrac{ 1 }{n^{3/2} }
  2. ∑n=1∞4n2\displaystyle\sum_{n=1}^{\infty} \dfrac{ 4 }{n^2} · ∑n=1∞4n3/2\displaystyle\sum_{n=1}^{\infty} \dfrac{ 4 }{n^{3/2} }

B

  1. It converges: the integral of 7xe^(−x²) from 1 to ∞ is finite

C

  1. 15\frac{1}{5}
  2. 11

D

  1. ∑n=1∞1n2\displaystyle\sum_{n=1}^{\infty} \dfrac{ 1 }{n^2} · ∑n=1∞1n3/2\displaystyle\sum_{n=1}^{\infty} \dfrac{ 1 }{n^{3/2} }
  2. ∑n=1∞4n2\displaystyle\sum_{n=1}^{\infty} \dfrac{ 4 }{n^2} · ∑n=1∞4n3/2\displaystyle\sum_{n=1}^{\infty} \dfrac{ 4 }{n^{3/2} }
  3. ∑n=1∞8n2\displaystyle\sum_{n=1}^{\infty} \dfrac{ 8 }{n^2} · ∑n=1∞8n3/2\displaystyle\sum_{n=1}^{\infty} \dfrac{ 8 }{n^{3/2} }

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