Leta(N) be such that (a(N)) is a decreasing sequence of strictly positive N=1 Numbers. IF S(N) denotes the Nth partial sum, show (by grouping the terms IN S(2) in two different ways) that: (a(1) + Za(2)+...+ 2a (2")) ≤ (a (1) + Za(z) + ... + 2Na (2N-1)) + a (2"). Use these inequalities to show that a(N) converges if and only if {2^a (2") converges. Use the Cauchy Condensation Test. N=1

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Let \(\sum_{N=1}^{\infty} a(N)\) be such that \(a(N)\) is a decreasing sequence of strictly positive numbers. If \(s(N)\) denotes the \(N\)th partial sum, show (by grouping the terms in \(s(2^N)\) in two different ways) that:

\[
\frac{1}{2}(a(1) + 2a(2) + \ldots + 2^N a(2^N)) \leq (a(1) + 2a(2) + \ldots + 2^{N-1} a(2^{N-1})) + a(2^N)
\]

Use these inequalities to show that \(\sum_{N=1}^{\infty} a(N)\) converges if and only if \(\sum_{N=1}^{\infty} 2^N a(2^N)\) converges. Use the Cauchy Condensation Test.
Transcribed Image Text:Let \(\sum_{N=1}^{\infty} a(N)\) be such that \(a(N)\) is a decreasing sequence of strictly positive numbers. If \(s(N)\) denotes the \(N\)th partial sum, show (by grouping the terms in \(s(2^N)\) in two different ways) that: \[ \frac{1}{2}(a(1) + 2a(2) + \ldots + 2^N a(2^N)) \leq (a(1) + 2a(2) + \ldots + 2^{N-1} a(2^{N-1})) + a(2^N) \] Use these inequalities to show that \(\sum_{N=1}^{\infty} a(N)\) converges if and only if \(\sum_{N=1}^{\infty} 2^N a(2^N)\) converges. Use the Cauchy Condensation Test.
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