let-R-and-a-n-k-1-n-sin-k-n-k-Find-lim-n-a-n-

Question Number 80332 by ~blr237~ last updated on 02/Feb/20

l e t α \in R a n d a_{n} = \underset{k = 1}{\sum^{n}} \frac{s i n (k α)}{n + k}

F i n d \lim_{n \to \infty} a_{n}

Commented by Rio Michael last updated on 03/Feb/20

a_{n} = \frac{\sin α}{n} + \frac{\sin 2 α}{n + 2} + \frac{\sin 3 α}{n + 3} + \dots

\lim_{n \to \infty} a_{n} = \lim_{n \to \infty} (\frac{\sin α}{n} + \frac{\sin 2 α}{n + 2} + \frac{\sin 3 α}{n + 3} + \dots)

For small angles \sin α = α

\Rightarrow \lim_{n \to \infty} a_{n} = \lim_{n \to \infty} (\frac{α}{n} + \frac{\sin 2 α}{n + 2} + \dots) = 0