Convergence Tests of Infinite Series

Convergence tests are methods for testing whether or not an infinite series converges to a finite value. This post is derived from my notes for MA101.

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Positive Term Series

Integral Test

Let $ \sum a_n $ be a positive term series, and let $ a_n = f(n) $ such that $ f(n) $ decreases as $ n $ increases. Then $ \sum a_n $ converges or diverges if $ \int_1^\infty f(x) dx $ is finite or infinite respectively.

p-Series Test

Let $ \sum a_n $ be a positive term series given by $a_n = \frac1{n^p} $. Then, $ \sum a_n $ is convergent if $ p \gt 1 $, and divergent if $ p \leq 1 $.

Comparison Test

Let $\sum a_n $ be a positive term series, then:

  1. $ \sum a_n $ is convergent if $ \sum b_n $ is another convergent series with $ a_n \leq b_n $.
  2. $ \sum a_n $ is divergent if $ \sum d_n $ is another divergent series with $ a_n \geq d_n $.
Limit Comparison Test

Let $ \sum a_n $ and $\sum b_n $ be two positive term series.

  1. If $ \lim_{n \rightarrow \infty} \frac{a_n}{b_n} $ is a finite and non-zero positive quantity, then $ \sum a_n $ and $\sum b_n$ will converge and diverge together.
  2. If $ \lim_{n \rightarrow \infty} \frac{a_n}{b_n} = 0 $ and $\sum b_n $ is convergent, then $ \sum a_n $ is also convergent.
  3. If $\lim_{n \rightarrow \infty} \frac{a_n}{b_n} = \infty $ and $ \sum b_n $ is divergent, then $ \sum a_n $ is also divergent.
D'Alembert's Ratio Test

Let $ \sum a_n $ be a positive term series, and let $ \lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_n} = r $.

  1. The series is convergent if $ r \lt 1 $.
  2. The series is divergent if $ r \gt 1 $, or if $r$ is infinite.
  3. The test fails if $r=1$.
Cauchy's Root Test

Let $ \sum a_n $ be a positive term series, and $ \lim_{n \rightarrow \infty} (a_n)^{\frac1n} = r $.

1 The series is convergent if $ r \lt 1 $.
2 The series is divergent if $ r \gt 1 $.
3 The test fails if $ r = 1 $.

Raabe's Test

Let $ \sum a_n $ be a positive term series, and $ \lim_{n \rightarrow \infty} n \left( \frac{a_n}{a_{n+1}}-1 \right) = k$.

  1. The series is convergent if $ k \gt 1 $.
  2. The series is divergent if $ k \lt 1 $.
  3. The test fails if $ k = 1 $.
Logarithmic Test

Let $ \sum a_n $ be a positive term series, and $ \lim_{n \rightarrow \infty} \log\left( \frac{a_n}{a_{n+1}} \right) = k$.

  1. The series is convergent if $ k \gt 1 $.
  2. The series is divergent if $ k \lt 1 $.
  3. The test fails if $ k = 1 $.

Alternating Series

Leibniz's Test

If the series $ \sum (-1)^n a_n $ is an alternating series, then the series is convergent if:

  1. Each term is numerically lesser than the preceeding term. ( $ |a_{n+1}| \lt |a_n| $ )
  2. $ \lim_{n \rightarrow \infty} a_n = 0 $.