2. By taking the limit N 0o, argue that if -12" azn converges then so does -1 an. (Hint: apply the monotone convergence theorem to the sequence of partial sums of the series an.) This is half of the proof that the Cauchy condensation test works. A similar argument can be used to show that if 12" azn diverges then so does En=1 an. You don't have to show this.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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2. By taking the limit \( N \to \infty \), argue that if \( \sum_{n=1}^{\infty} 2^n a_{2^n} \) converges then so does \( \sum_{n=1}^{\infty} a_n \). (Hint: apply the monotone convergence theorem to the sequence of partial sums of the series \( \sum_{n=1}^{\infty} a_n \).)

This is half of the proof that the Cauchy condensation test works. A similar argument can be used to show that if \( \sum_{n=1}^{\infty} 2^n a_{2^n} \) diverges then so does \( \sum_{n=1}^{\infty} a_n \). You don’t have to show this.
Transcribed Image Text:2. By taking the limit \( N \to \infty \), argue that if \( \sum_{n=1}^{\infty} 2^n a_{2^n} \) converges then so does \( \sum_{n=1}^{\infty} a_n \). (Hint: apply the monotone convergence theorem to the sequence of partial sums of the series \( \sum_{n=1}^{\infty} a_n \).) This is half of the proof that the Cauchy condensation test works. A similar argument can be used to show that if \( \sum_{n=1}^{\infty} 2^n a_{2^n} \) diverges then so does \( \sum_{n=1}^{\infty} a_n \). You don’t have to show this.
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