Aspire Faculty ID #3630 · Topic: NIMCET 2012 · Just now
NIMCET 2012

The value of $\displaystyle \lim_{n\to\infty} \frac{\pi}{n}\left[\sin\frac{\pi}{n}+\sin\frac{2\pi}{n}+\cdots+\sin\frac{(n-1)\pi}{n}\right]$ is:

Solution

This is a Riemann sum. 
 Rewrite the expression: 
 $\displaystyle \frac{\pi}{n}\sum_{k=1}^{n-1}\sin\left(\frac{k\pi}{n}\right)$ 
 Let $x_k = \frac{k\pi}{n}$ 
Then spacing $\Delta x = \frac{\pi}{n}$ 
 As $n\to\infty$, this becomes: 
 $\displaystyle \int_{0}^{\pi} \sin x , dx$ 
 Now evaluate: $\displaystyle \int_{0}^{\pi} \sin x  dx = [-\cos x]_{0}^{\pi}$ 
 $= (-\cos\pi) - (-\cos 0)$ 
 $= -(-1) - (-1)$ 
 $= 1 + 1 = 2$ 
 $\therefore$ the value of the limit is 2. 

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