Abel's Test

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


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


Proof


Source of Name

This entry was named for Niels Henrik Abel.


Sources