Choosing a convergence test

Warm-up

Three quick questions from earlier lessons.

Discover

$\displaystyle\sum \dfrac{n}{2n + 1}$ : before any clever test, what should you check first?

Whether the terms tend to 0
The ratio test

Key idea

A short checklist.

1.

a_n \not\to 0

: diverges. 2. Geometric or p-series: read off. 3. Alternating: alternating series test. 4. Factorials or

c^n

: ratio test. 5. Rational or root expressions: compare with a p-series. 6. Easy integral: integral test.

Worked example

Choose tests for three series.

\displaystyle\sum \dfrac{n!}{10^n}

: ratio test,

L = \infty

, diverges

Factorial.

\displaystyle\sum \dfrac{(-1)^n}{\ln(n + 1)}

: alternating series test, converges

Signs alternate, terms decrease to 0.

\displaystyle\sum \dfrac{1}{n^2 + n}

: comparison, converges

Leading terms 1/n².

Your turn, guided

Your turn: $\displaystyle\sum \left(\tfrac{2}{3}\right)^n$ is geometric with ratio…

r = \tfrac{2}{3}

|r| < 1: converges.

Practice ★

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 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

Watch out!

Practice ★★

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

Exit ticket

Three questions to check you have got it.

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

Summary card

Choosing a test

nth-term → known series → alternating → ratio → comparison → integral.

Watch out: a divergent absolute series does not make an alternating series diverge.

About this lesson