# limit problem...

• April 23rd 2009, 08:28 AM
Mathventure
limit problem...
Q:let fn(x)=n(sinx)^(2n+1)*cos(x)
then find the value of
{ limt n tends to inf of integral of fn(x) from 0 to pi/2}
-{integral of limit n tends to inf of fn(x) from 0 to pi/2}
• May 27th 2009, 06:48 AM
Media_Man
Fatou's Lemma
Define $f_n(x)=n\sin^{2n+1}(x)\cos(x)$

Find $\lim_{n\rightarrow\infty}\int_0^{\pi/2} f_n(x)dx - \int_0^{\pi/2} \lim_{n\rightarrow\infty}f_n(x)dx$

Integrating by simple substitution $\int n\sin^{2n+1}(x)\cos(x)dx=\frac{n}{2n+2}\sin^{2n+2} (x)$

So $\lim_{n\rightarrow\infty}\int_0^{\pi/2} f_n(x)dx=\lim_{n\rightarrow\infty}\frac{n}{2n+2}=\ frac12$

But $\lim_{n\rightarrow\infty}n\sin^{2n+1}(x)\cos(x)=0$, so your answer is zero.