# Abel's Test

## Theorem

Let $\displaystyle \sum a_n$ be a convergent real series.

Let $\sequence {b_n}$ be a decreasing sequence of positive real numbers.

Then the series $\displaystyle \sum a_n b_n$ is also convergent.

### Abel's Test for Uniform Convergence

Let $\sequence {\map {a_n} z}$ and $\sequence {\map {b_n} z}$ be sequences of complex functions on a compact set $K$.

Let $\sequence {\map {a_n} z}$ be such that:

$\sequence {\map {a_n} z}$ is bounded in $K$
$\ds \sum \size {\map {a_n} z - \map {a_{n + 1} } z}$ is convergent with a sum which is bounded in $K$
$\ds \sum \map {b_n} z$ is uniformly convergent in $K$.

Then $\ds \sum \map {a_n} z \map {b_n} z$ is uniformly convergent on $K$.

## Source of Name

This entry was named for Niels Henrik Abel.