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Calculus II · Sequences & Series

Convergence Tests — Practice Problems and Worked Solutions

Choose comparison, ratio, or the nth-term test with intent.

The formulas you need here

Ratio test

L = n \to \infty lim \frac{a _{n + 1}}{a _{n}}; L < 1 \Rightarrow converges

What it says: Compare each term with the one before; if the ratio settles below 1, the sum converges absolutely.
When to reach for it: Factorials, or a variable in an exponent — anything where the ratio simplifies dramatically.
Watch out: L = 1 is a genuine no-answer. Switch to comparison or the integral test instead of guessing.

\sum \frac{2 ^{n}}{n !} : L = lim \frac{2}{n + 1} = 0 < 1

Factorials always beat exponentials, so this converges.

Alternating series test

\sum (- 1)^{n} b_{n} converges if b_{n} ↓ 0

What it says: Alternating signs with terms that decrease steadily to zero always converge.
When to reach for it: A (−1)ⁿ or (−1)^{n+1} out front, once the divergence test has come back clean.
Watch out: Convergence here can be conditional. Check Σ|aₙ| separately if the question asks about absolute convergence.

\sum \frac{( - 1 ) ^{n}}{n} converges, \sum \frac{1}{n} does not

The textbook example of conditional convergence.

16 worked problems

Every step is generated and checked by a computer algebra system.

1.
Determine whether the series converges.
$n = 1 \sum \infty \frac{2 ^{n}}{n !}$
Answer: $Converges$
Step-by-step solution
1. $Ratio test: L = 0$
2. $Converges$
2.
Determine whether the series converges.
$n = 1 \sum \infty \frac{n !}{3 ^{n}}$
Answer: $Diverges$
Step-by-step solution
1. $Terms eventually grow; a_{n} \neq \to 0$
2. $Diverges$
3.
Determine whether the series converges.
$n = 1 \sum \infty \frac{n}{2 ^{n}}$
Answer: $Converges$
Step-by-step solution
1. $Ratio test: L = \frac{1}{2}$
2. $Converges$
4.
Determine whether the series converges.
$n = 1 \sum \infty \frac{2 ^{n}}{n ^{3}}$
Answer: $Diverges$
Step-by-step solution
1. $a_{n} \neq \to 0$
2. $Diverges$
5.
Determine whether the series converges.
$n = 1 \sum \infty \frac{1}{n ^{2} + 1}$
Answer: $Converges$
Step-by-step solution
1. $Compare with \frac{1}{n ^{2}}$
2. $Converges$
6.
Determine whether the series converges.
$n = 1 \sum \infty \frac{1}{n + 1}$
Answer: $Diverges$
Step-by-step solution
1. $Limit compare with \frac{1}{n}$
2. $Diverges$
7.
Determine whether the series converges.
$n = 1 \sum \infty \frac{3 ^{n}}{n !}$
Answer: $Converges$
Step-by-step solution
1. $Ratio test: L = 0$
2. $Converges$
8.
Determine whether the series converges.
$n = 1 \sum \infty \frac{n !}{2 ^{n}}$
Answer: $Diverges$
Step-by-step solution
1. $Terms eventually grow; a_{n} \neq \to 0$
2. $Diverges$
9.
Determine whether the series converges.
$n = 1 \sum \infty \frac{n}{3 ^{n}}$
Answer: $Converges$
Step-by-step solution
1. $Ratio test: L = \frac{1}{3}$
2. $Converges$
10.
Determine whether the series converges.
$n = 1 \sum \infty \frac{3 ^{n}}{n ^{2}}$
Answer: $Diverges$
Step-by-step solution
1. $a_{n} \neq \to 0$
2. $Diverges$
11.
Determine whether the series converges.
$n = 1 \sum \infty \frac{1}{n ^{3} + 2}$
Answer: $Converges$
Step-by-step solution
1. $Compare with \frac{1}{n ^{3}}$
2. $Converges$
12.
Determine whether the series converges.
$n = 1 \sum \infty \frac{1}{3 n}$
Answer: $Diverges$
Step-by-step solution
1. $p -series with p = \frac{1}{3} \leq 1$
2. $Diverges$
13.
Determine whether the series converges.
$n = 1 \sum \infty \frac{n ^{2}}{n !}$
Answer: $Converges$
Step-by-step solution
1. $Ratio test: L = 0$
2. $Converges$
14.
Determine whether the series converges.
$n = 1 \sum \infty \frac{n + 1}{n}$
Answer: $Diverges$
Step-by-step solution
1. $a_{n} \to 1 \neq = 0$
2. $Diverges$
15.
Determine whether the series converges.
$n = 1 \sum \infty \frac{1}{n ln n}$
Answer: $Diverges$
Step-by-step solution
1. $Integral test diverges$
2. $Diverges$
16.
Determine whether the series converges.
$n = 1 \sum \infty \frac{1}{2 ^{n}}$
Answer: $Converges$
Step-by-step solution
1. $Geometric with r = \frac{1}{2}$
2. $Converges$

The formulas you need here

Ratio test

Alternating series test

16 worked problems

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