Let an=∫ limits0π / 2(1- sin t)n sin 2 t d t then undersetn arrow ∞ textLim displaystyle∑n=1n (an/n) is equal to

Question

Let an=∫ limits0π / 2(1- sin t)n sin 2 t d t then undersetn arrow ∞ textLim displaystyle∑n=1n (an/n) is equal to (A) 1 / 2 (B) 1 (C) 4 / 3 (D) 3 / 2

Tardigrade Admin · Accepted Answer

an=∫ limits0π / 2(1- sin t)n sin 2 t d t Let 1- sin t = u ⇒- cos t dt = du an=2 ∫ limits01 un(1-u) d u=2(∫ limits01 un d u-∫ limits01 un+1 d u)=2((1/n+1)-(1/n+2)) hence (an/n)=2((1/n(n+1))-(1/n(n+2))) undersetn arrow ∞ textLim displaystyle∑n=1n (an/n) =2(∑((1/n)-(1/n+1))-(1/2) displaystyle∑((1/n)-(1/n+2)))=2 ∑n=1n((1/n)-(1/n+1))- displaystyle∑n=1n((1/n)-(1/n+2)) =2(1)-[(1-(1/3))+((1/2)-(1/4))+((1/3)-(1/5))+ ldots . .]=2-(3/2)=(1/2)

Q. Let $a_{n} = 0 \int π /2 (1 - sin t)^{n} sin 2 t d t$ then $n \to \infty Lim n = 1 \sum n \frac{a _{n}}{n}$ is equal to

$1/2$

1

$4/3$

$3/2$

Q. Let an​=0∫π/2​(1−sint)nsin2tdt then n→∞Lim​n=1∑n​nan​​ is equal to

1/2

1

4/3

3/2

Q. Let $a_{n} = 0 \int π /2 (1 - sin t)^{n} sin 2 t d t$ then $n \to \infty Lim n = 1 \sum n \frac{a _{n}}{n}$ is equal to

$1/2$

$4/3$

$3/2$