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.

This content is also available in PDF format.

### 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. $\lim_{n \rightarrow \infty} a_n = 0$.