Series Basics

Theory

A series is the sum of the terms of a sequence.
If $\{a_n\}$ is a sequence, then the corresponding series is

\sum_{n=1}^{\infty} a_n = a_1 + a_2 + a_3 + \cdots

The $n$ -th partial sum of a series is

S_n = \sum_{k=1}^{n} a_k.

The series $\sum a_n$ converges if

\lim_{n\to\infty} S_n = S \quad (\text{finite}).

Otherwise, the series diverges.

\lim_{n\to\infty} a_n \neq 0,

then the series $\sum a_n$ diverges.

Important:
If $\lim a_n = 0$ , the test is inconclusive.

A series is geometric if

\sum_{n=1}^{\infty} ar^{\,n-1}

for constants $a$ and $r$ .

S=\frac{a}{1-r}.

A series is telescoping if its partial sums cancel:

S_n = (a_1-a_2)+(a_2-a_3)+\cdots+(a_n-a_{n+1}).

Most terms cancel, leaving only a few boundary terms.
Take the limit of $S_n$ to determine convergence.

\sum \ln(a_n),

then

S_n = \ln\!\left(\prod a_n\right).

If $S_n \to -\infty$ , the series diverges.

For rational expressions involving $n$ :

For expressions of the form

\left(1+\frac{c}{n}\right)^n,

use the limit

\lim_{n\to\infty}\left(1+\frac{c}{n}\right)^n = e^{c}.

The harmonic series is

\sum_{n=1}^{\infty} \frac{1}{n} = 1 + \frac12 + \frac13 + \frac14 + \cdots

Fact:
The harmonic series diverges.

A p-series is a series of the form

\sum_{n=1}^{\infty} \frac{1}{n^p},

where $p$ is a real number.

Examples:

\sum \frac{1}{n^2} \quad \text{converges}, \qquad \sum \frac{1}{n} \quad \text{diverges}

The harmonic series is the boundary case of the p-series when $p=1$ . Even though

\lim_{n\to\infty} \frac{1}{n} = 0,

the series still diverges.