Abel's Test

Theorem

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

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

Then the series $\ds \sum a_n b_n$ 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 $K$.

Proof

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Source of Name

This entry was named for Niels Henrik Abel.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Entry: Abel's test: 1.
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Entry: Abel's test
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Entry: Abel's test
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Entry: Abel's test