Abel's Test

Theorem

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

Let $\sequence {a_n}$ 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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Let $b_0 = 0$.

Let $B_N = \ds \sum_{k \mathop = 0}^N b_k$.

Then:

$\forall n \ge 1: b_n = B_n − B_{n − 1}$

From Abel's Lemma:

$\ds \sum_{k \mathop = 1}^N a_k b_k = \sum_{k \mathop = 1}^{N - 1} B_k \paren {a_k - a_{k + 1} } + a_N B_N$

By the Monotone Convergence Theorem:

$\sequence {a_n}$ converges

$\sequence {B_N}$ converges since $\ds \sum b_n$ converges.

Hence the second addend $a_N B_N$ converges.

It remains to prove the first addend $\ds \sum B_k \paren {a_k − a_{k + 1} }$ converges.

Since $\sequence {B_N}$ converges, $\size {B_N} \le M$ for some $M \in \N$.

$\ds \sum_1^N \size {B_k \paren {a_k -a_{k + 1} } }$

$\le$

$\ds M \sum_1^N \size {a_k - a_{k + 1} }$

$\ds $

$=$

$\ds M \size {a_1 - a_{N + 1} }$

as $\sequence {a_n}$ is decreasing

$\ds $

$\to$

$\ds M \size {a_1 - a}$

as $a_k \to a$

Hence $\ds \sum B_k \paren {a_k − a_{k + 1} }$ converges absolutely.

Hence $\ds \sum a_n b_n$ converges.

$\blacksquare$

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): Abel's test: 1.
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Abel's test
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Abel's test
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Abel's test
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Abel's test