Partial sum

$\displaystyle \sum_{n=1}^T a_n$ is called the $T$th partial sum of the series $\displaystyle \sum_{n=1}^\infty a_n$. A series converges if the sequence of partial sums converges.

$n$th term test

If $\displaystyle \lim_{n \rightarrow \infty} a_n \ne 0$, then the series $\displaystyle \sum_{n=0}^\infty a_n$ diverges. Can be used to test for divergence (but not convergence). ie The converse is not always true, but the contra-positive is.

Integral test

$$\begin{aligned} \sum_{n=m}^\infty f(n) \hskip1em \text{converges} \hskip2em \Leftrightarrow \hskip2em \int_{m}^\infty f(x) \, dx \hskip1em \text{converges} \\ \end{aligned}$$

Condition: $f(x)$ is a positive decreasing function.

p-series test

$$\begin{aligned} \sum_{n=1}^\infty \frac{1}{n^p} \hskip1em \text{converges if and only if } p > 1 \\ \end{aligned}$$

Comparison test

Let $a_n$ and $b_n$ be positive sequences such that $b_n > a_n$ for all $n$. It follows that if $\sum b_n$ converges, then so does $\sum a_n$, and if $\sum a_n$ diverges, then so does $\sum b_n$.

Limit test

Let $a_n$ and $b_n$ be positive series. If $\displaystyle \lim_{n \rightarrow \infty} \frac{a_n}{b_n} = L$ and $0 < L < \infty$, then the series $\sum a_n$ and $\sum b_n$ either both converge or both diverge.

Root test

Given a series $\sum a_n$ with all $a_n > 0$, let $\displaystyle \rho = \lim_{n \rightarrow \infty} \sqrt[n]{a_n}$. If $\rho$ < 1, then $\sum a_n$ converges. If $\rho$ > 1, then $\sum a_n$ diverges. If $\rho$ = 1, then the test is inconclusive. Based on the idea of "eventual geometric series".

Ratio test

Given a series $\sum a_n$ with all $a_n > 0$, let $\displaystyle \rho = \lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_n}$. If $\rho$ < 1, then $\sum a_n$ converges. If $\rho$ > 1, then $\sum a_n$ diverges. If $\rho$ = 1, then the test is inconclusive. Work best when exponential or factorial factor is involved - not so with ratio of polynomials.

Summary of methods for a positive series

Given a series $\sum a_n$, where $a_n > 0$ for all $n$:

1. Do the terms go to 0 ? If $\displaystyle \lim_{n \rightarrow \infty} a_n \ne 0$, then by the nth term test, the series diverges. (If $\displaystyle \lim_{n \rightarrow \infty} a_n = 0$, then the test is inconclusive).
2. Does $a_n$ involve exponential functions like $c^n$ where $c$ is constant ? Does $a_n$ involve factorial ? Then the ratio test should be used.
3. Is $a_n$ of the form $(b_n)^n$for some sequence $b_n$ ? Then use the root test.
4. Does ignoring lower order terms make $a_n$ look like a familiar series (e.g. p-series or geometric series) ? Then use the comparison test or the limit test.
5. Does the sequence $a_n$ look easy to integrate ? Then use the integral test.

Alternating Series test

An alternating series $\sum (-1)^n b_n$ with decreasing terms converges if and only if $\displaystyle \lim_{n \rightarrow \infty} b_n = 0$. (Note when to examine the asymptotic value of the $n$th term vs. examining the p-value of an integral.)

Conditional vs Absolute Convergence

$\displaystyle \sum a_n$ converges absolutely if $\displaystyle \sum \lvert a_n \rvert$ converges. $\displaystyle \sum a_n$ converges conditionally if $\displaystyle \sum \lvert a_n \rvert$ diverges but $\displaystyle \sum a_n$ converges. In other words, a series is conditionally convergent if it is convergent but not absolutely convergent.

Absolute Convergence test

If a series converges absolutely, then it always converges. (But this is not necessarily true if it converges conditionally.)

Interesting notes

• $\displaystyle \lim_{n \rightarrow \infty} \sqrt[n]{n} = 1$

  $$\begin{aligned} y &= \lim_{n \rightarrow \infty} \sqrt[n]{n} = \lim_{n \rightarrow \infty} n^\frac{1}{n} \\ ln(y) &= \lim_{n \rightarrow \infty} \frac{1}{n} ln(n) = 0 \\ \therefore y &= 1 \\ \end{aligned}$$

• $\displaystyle \lim_{n \rightarrow \infty} (1 + \frac{1}{n})^n = e = \sum_{n=0}^\infty \frac{1}{n!}$
• $\displaystyle \lim_{n \rightarrow \infty} (1 + \frac{a}{n})^n = e^a = \sum_{n=0}^\infty \frac{1}{n!} a^n$
• $\displaystyle 0 < ln(2) < 1$