Prove or disprove: If the sequence I_n converges, then ∫ f(x) converges

by RoRi

March 31, 2016

The following function $f$ is defined for all $x \geq 1$ , and $n$ is a positive integer. Prove or provide a counterexample to the following statement.

The convergence of the sequence $\{ I_n \}$ implies the convergence of the integral

$\int_1^{\infty} f(x) \, dx.$

Incomplete.

2 comments

July 30, 2021 at 11:09 am

Evangelos says:

Last post got spam filtered. I’ll keep it simple here:
Counterexample. $f(t) &= \cos{2\pi t)$

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July 30, 2021 at 11:05 am

Evangelos says:

Counterexample. Let $f(t) &= \cos{2 \pi t}$ . Then, for all positive integers $n$ ,

$\begin{align*} I_{n} &= 0 \end{align*}$

But $\lim_{x \to \infty} \int_{1}^{x} f(t) \,dt$ diverges.

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Stumbling Robot

Prove or disprove: If the sequence I_n converges, then ∫ f(x) converges

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