1

I have been struggling with the following problem from "Problems in Real Analysis: Advanced Calculus on the Real Axis":

3.4.10. Let $f$ be a positive strictly increasing function on $[1,+\infty),$ with $f(x) \rightarrow+\infty$ as $x \rightarrow+\infty$. Prove that the series $$\sum_{n=1}^{\infty} 1 / f(n)$$ and $$\sum_{n=1}^{\infty} n^{-2} f^{-1}(n)$$ converge only simultaneously.

I tried using the limit comparison test, but couldn't get to a limit I could prove exists.

vitamin d
  • 5,783
exhaustmanifold
  • 23
  • 2
    Would be useful to check how they define the inverse function $f^{-1}(y)$ as the conditions given are not sufficient to guarantee that this inverse function exists. Otherwise, I think the idea is to set $g(x)=\frac{1}{f(x)}$ and then compare the first sum with $\int_1^{\infty}g(x)dx$ and the second sum with $\int_0^1 g^{-1}(y)dy$ - the two integrals are the "area under the curve" $y=g(x)$ expressed in two different ways... My calculus is a bit rusty, but suppose I can give it a try and maybe post a bit later if I've got a full proof. You may try to complete it yourself. –  Feb 27 '21 at 17:39

0 Answers0