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Study Stuff for MCA Examinations - NIMCET
Study Stuff for MCA Examinations - NIMCET
Previous Year Question (PYQs)
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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