series

Series

Let $(a_n{)}_{n\in \mathbb{N}}$ be a sequence. The $n$-th partial sum is $s_n:={\sum }_{k=1}^n{a}_k$.
The series ${\sum }_{k=1}^{\infty }{a}_k$ is the limit of the partial sums:

$$\sum _{k=1}^{\infty }{a}_k:=\underset{n\to \infty }{\lim }{s}_n=\underset{n\to \infty }{\lim }\sum _{k=1}^n{a}_k$$

If this limit exists (finite), the series converges and we call the sequence $(a_n)$ summable.
If $s_n\to \pm \infty$, the series is definitely divergent.
Otherwise (partial sums keep bouncing, e.g. $\sum (-1{)}^k$) the series is divergent.

A series is just a sequence of partial sums. All the tools for sequences (convergence, boundedness, Cauchy, …) apply to $(s_n)$.

Necessary condition for convergence

$$\sum a_k\text{ converges}⟹a_k\to 0$$

If $a_k\nrightarrow 0$, the series diverges.

The converse is false: $a_k=1/k\to 0$ but $\sum 1/k=\infty$ (see harmonic series).

Calculation rules

If $\sum a_k$ and $\sum b_k$ both converge, then:

$$\sum (a_k+b_k)=\sum a_k+\sum b_k,\sum c\cdot a_k=c\cdot \sum a_k$$

Non-negative terms

If $a_k\ge 0$ for all $k$, then the partial sums $s_n$ are non-decreasing. By the monotonicity principle, the series converges iff the partial sums are bounded.

See also: geometric series, harmonic series, telescoping series