I_n = _0^/4 ^nx,dx then _n n[I_n+I_n+2] equals:

Question

$I_n = \displaystyle\int_0^{\pi/4} 	an^nx\,dx$ then $\displaystyle\lim_{n	o\infty} n[I_n+I_{n+2}]$ equals:

Accepted Answer

$I_n+I_{n+2} = \int_0^{\pi/4} 	an^nx(1+	an^2x)\,dx = \int_0^{\pi/4}	an^nx\sec^2x\,dx = \left[\dfrac{	an^{n+1}x}{n+1}ight]_0^{\pi/4} = \dfrac{1}{n+1}$

$\lim_{n	o\infty} n[I_n+I_{n+2}] = \lim_{n	o\infty}\dfrac{n}{n+1} = \lim_{n	o\infty}\dfrac{n}{n\left(1+\dfrac1night)} = 1$

Solutions · 1