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 b0 = 0, $B_N=\sum_{k=0}^N b_k$. Then bn = Bn − Bn − 1, n ≥ 1, hence \begin{align*} \sum_{k=1}^N a_kb_k &= \sum_{k=1}^N a_k(B_k-B_{k-1}) \\ &=B_1(a_1-a_2) +B_2(a_2-a_3)+\dots B_{N-1}(a_{N-1}-a_N)+a_NB_N\\ &=\sum_{k=1}^{N-1} B_k(a_k-a_{k+1})+a_NB_N \end{align*} [This identity is Abel's Lemma/Formulation 2.]
By Monotone Convergence Theorem {an} converges; and {BN} converges since ∑bn converges. Hence the second addend aNBN converges.
It remains to prove the first addend ∑Bk(ak−ak+1) converges.
Since $\sequence{B_N}$ converges, $\bigsize{B_N}\le M$ for some $M$.
\(\ds \sum_1^N\bigsize{B_k (a_k-a_{k+1})}\) | \(\le\) | \(\ds M \sum_1^N \bigsize{a_k-a_{k+1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds M\bigsize{a_1-a_{N+1} }\) | since {an} is decreasing | |||||||||||
\(\ds \) | \(\to\) | \(\ds M \bigsize{a_1-a}\) | since ak → a. |
Hence ∑Bk(ak − ak+1) converges absolutely.
Hence $\ds \sum a_n b_n$ converges.
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