Numerical series

Definition
We call a numerical series and represent it with the symbol

\sum_{n = 0}^{\infty} a_{n}

the pair of sequences $((a_{n})_{n}, (s_{n})_{n})$ , where:

$(a_{n})_{n}$ is a sequence of complex numbers, whose terms $a_{n}$ are called the terms of the series.
$(s_{n})_{n}$ is the sequence called the sequence of partial sums of the series such that, for each $n \in N$ :

s_{n} = a_{0} + a_{1} + \dots + a_{n} = \sum_{k = 0}^{n} a_{k}

Definition
We say that the series

\sum_{n = 0}^{\infty} a_{n}

is convergent if the sequence of partial sums $(s_{n})_{n}$ converges. In that case, the sum of the series is defined as the limit

\sum_{n = 0}^{\infty} a_{n} = lim_{n \to \infty} s_{n}

Definition
We say that the series

\sum_{n = 0}^{\infty} a_{n}

converges absolutely if the series of the moduli

\sum_{n = 0}^{\infty} | a_{n} |

converges.

Proposition. Convergence Criteria
The following criteria for the convergence of series of complex numbers hold:

Cauchy Criterion. The series

\sum_{n = 0}^{\infty} a_{n}

is convergent if and only if for every $ε > 0$ there exists $n_{0} \in N$ such that for all $p \geq n_{0}$ and $q \geq 0$ :

| \sum_{k = p}^{p + q} a_{k} | < ε

Comparison Criterion. Let

\sum_{n = 0}^{\infty} a_{n}, \sum_{n = 0}^{\infty} b_{n}

be two series of complex numbers. Suppose that

| a_{n} | \leq | b_{n} | for n \geq 0

Then if $\sum_{n = 0}^{\infty} | b_{n} |$ converges, then $\sum_{n = 0}^{\infty} | a_{n} |$ also converges.
3. Root Test. Let $\sum_{n = 0}^{\infty} a_{n}$ be a series and suppose the limit

lim_{n \to \infty} \sqrt[n]{| a_{n} |} = ρ

exists. Then, if $ρ < 1$ , the series converges absolutely.

Ratio Test. Let $\sum_{n = 0}^{\infty} a_{n}$ be a series and suppose the limit

lim_{n \to \infty} | \frac{a_{n + 1}}{a_{n}} | = ρ

exists. Then, if $ρ < 1$ , the series converges absolutely.

Related: series of functions.