Series

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

Convergence

If is converging:

Absolutely Converging

is absolutely converging iff .

Proof Hint

Use this inequality:

Theorem

A series is absolutely converging to iff rearranged series of converges to .

Conditionally Converging

is condtionally converging iff:

Theorem

Suppose is a conditionally converging series. Then:

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

Terms limit to 0

The converse is known as the divergence test:

Grouping

Suppose is a given series. If is formed by grouping a finite number of adjacent terms , then is a grouping of the given series.

Rearrangement

Suppose is a given series. If there is a bijection sequence such that , then is a rearrangement of the given series.