Does $\int_1^\infty (f(x))^2dx$ converge if $\int_1^\infty f(x)dx$ converges for positive continuous $f$?

Asked Jan 13 '24 at 06:17

Active Jan 13 '24 at 06:49

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Let $f:\mathbb{R}\mapsto \mathbb{R}$ be a positive continuous function. If $\int_1^\infty f(x)dx$ converges, is it necessarily true that $\int_1^\infty (f(x))^2dx$ converges?

If it is true, explain why. If it is not true, explain with a counterexample.

I have tried many ways but failed to solve. Please assist me.

edited Jan 13 '24 at 06:49

Benjamin Wang

4,667

asked Jan 13 '24 at 06:17

Gasp willson

3

What have you tried? It is hard to help you without that information. – Benjamin Wang Jan 13 '24 at 06:47
Cauchy-Schwartz-Bunjakowski? Hölder? – Matcha Latte Jan 13 '24 at 06:54
Have you developed any claims about the behavior of $f$ as $x \rightarrow \infty$? – Steven Cripe Jan 13 '24 at 07:33
1

[Find a continuous function $f:1,\infty)\to\Bbb R $ such that $f(x) >0 $, $\int_1^\infty f(x),dx $ converges and $\int_1^\infty f(x)^2,dx$ diverges – Sangchul Lee Jan 13 '24 at 07:36

Does $\int_1^\infty (f(x))^2dx$ converge if $\int_1^\infty f(x)dx$ converges for positive continuous $f$?

0 Answers0