Series

Let $(u_n)$ be a sequence, and a series (a new sequence) can be defined from it such that:

Convergence

If $(s_n)$ is converging:

Absolutely Converging

$\sum_{k=1}^{n}{u_k}$ is absolutely converging iff $\sum_{k=1}^{n}{\lvert u_k \rvert}\text{ is converging}$.

A series $s_n$ is absolutely converging to $s$ iff rearranged series of $s_n$ converges to $s$.

Conditionally Converging

$\sum_{k=1}^{n}{u_k}$ is condtionally converging iff:

Suppose $s_n$ is a conditionally converging series. Then:

1. Sum of all the positive terms limits to $\infty$
2. Sum of all the negative terms limits to $-\infty$
3. $s_n$ can be rearranged to have the sum:
   • Any real number $x$
   • $\infty$
   • $-\infty$
   • Does not exist

Terms Limit to 0

The converse is known as the divergence test:

Grouping

Suppose $\sum u_k$ is a given series. If $v_n$ is formed by grouping a finite number of adjacent terms $u_k$, then $\sum v_k$ is a grouping of the given series.

Rearrangement

Suppose $\sum u_k$ is a given series. If there is a bijection sequence $k_n$ such that $v_n = u_{k_n}$, then $\sum v_n$ is a rearrangement of the given series.