# Thread: This is a problem under Riemann integral. Please help how to proceed.

1. ## This is a problem under Riemann integral. Please help how to proceed.

lim n-infinity (1/n)(sin pi/n +sin 2pi/n + sin 3pi/n +...+sin npi/n) ?

2. ## Re: This is a problem under Riemann integral. Please help how to proceed.

Recall:

$\int_a^b f(x)\,dx=\lim_{n\to\infty}\left[\frac{b-a}{n}\sum_{k=1}^nf\left(a+k\frac{b-a}{n} \right) \right]$

We are asked to evaluate:

$\lim_{n\to\infty}\left[\frac{1}{n}\sum_{k=1}^n\sin\left(\frac{k\pi}{n} \right) \right]$

So, let:

$a=0,b=1,f(x)=\sin(\pi x)$

Can you finish?

3. ## Re: This is a problem under Riemann integral. Please help how to proceed.

Originally Posted by anushua
lim n-infinity (1/n)(sin pi/n +sin 2pi/n + sin 3pi/n +...+sin npi/n) ?