Series of Harmonic sum

Asked Nov 13 '21 at 12:51

Active Nov 13 '21 at 13:06

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Let $H_n=\sum_{k=1}^n \frac{1}{k}$, then find all real numbers $p$ such that the series $\sum_{n\in\mathbb{N}} \frac{1}{H_n^p}$ is convergent.

My attempt is: There exist a subseqence of natural numbers $(n_k)$ such that $H_{n_k}>k$. Hence for $p>1$ we have sub series of the main series which is convergent.

But I can not conclude anything for the main series except $p<0$. Please help me to solve this.

edited Nov 13 '21 at 12:57

RAHUL

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asked Nov 13 '21 at 12:51

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What does it matter that the series contains a convergent subsequence? $\sum \frac 1n$ has many convergent subsequences, but it still diverges. – lulu Nov 13 '21 at 12:56
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That "subseries" proves nothing: If $a_n$ is any sequence tending to zero there is a subsequence $a_{n_k}$ such that $\sum_k a_{n_k}$ is convergent... – David C. Ullrich Nov 13 '21 at 12:56
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Have you tried using the fact that $H_n \sim \log n$ as $n\to \infty$? – Minus One-Twelfth Nov 13 '21 at 12:57
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You need to start with an estimate of approximately how large $H_n$ is. – David C. Ullrich Nov 13 '21 at 12:57

Series of Harmonic sum

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