Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

How can we show that if $f:\mathbb{N} \rightarrow \mathbb{N}$ with $f(n+1) \geq f(n)>1$ for all $n\ge 1$ and $\sum_{n = 1}^\infty \frac{1}{f(n)} = \infty$, then for all integers $k>1$ we have $\sum_{n = 1}^\infty \frac{1}{f(kn)} = \infty$?

share|cite|improve this question
If the sum is finite, then so is $\sum \frac{1}{f(kn+r)}$ where $0\le r \le k-1$. – André Nicolas Nov 14 '11 at 7:43

1 Answer 1

up vote 5 down vote accepted

$$ \frac{k}{f(kn)}\geqslant\frac1{f(kn+1)}+\frac1{f(kn+2)}+\cdots+\frac1{f(kn+k)}. $$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.