1
$\begingroup$

How can we show $\lim_{n\rightarrow\infty}\int_X\cos(nx) \rightarrow 0\;$?

(X can be any set)

$\endgroup$

3 Answers 3

1
$\begingroup$

$$\int_X\cos nx\,dx=\left.\frac{1}{n}\sin nx\right|_X\xrightarrow [n\to\infty]{} 0$$

as $\,\sin nx\,$ is bounded on $\,X\subset \Bbb R\,$ (why?)

$\endgroup$
1
$\begingroup$

Begin by doing it for a closed bounded interval. Then exploit the basic properties of limits.

$\endgroup$
1
$\begingroup$

Hint Check Riemann–Lebesgue lemma.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .