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[parent] proof of Abel's test for convergence (Proof)

Let $ b$ be the limit of $ \{b_n\}$ and let $ d_n=b_n-b$ when $ \{b_n\}$ is decreasing and $ d_n=b-b_n$ when $ \{b_n\}$ is increasing. By Dirichlet's convergence test, $ \sum a_nd_n$ is convergent and so is $ \sum a_nb_n = \sum a_n(b\pm d_n) = b\sum a_n \pm \sum a_nd_n$.



"proof of Abel's test for convergence" is owned by lieven.
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Cross-references: convergent, Dirichlet's convergence test, increasing, decreasing, limit

This is version 1 of proof of Abel's test for convergence, born on 2002-12-27.
Object id is 3845, canonical name is ProofOfAbelsTestForConvergence.
Accessed 5513 times total.

Classification:
AMS MSC: 40A05 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences)
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