Find lim_(x→∞) Σ_(k=1) ^(n) (1/n)sin((2π)/n)


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Question Number 31707 by gunawan last updated on 13/Mar/18

$Find \lim_{x \to \infty} \underset{k = 1}{\overset{n}{Σ}} \frac{1}{n} \sin \frac{2 π}{n}$
Commented by abdo imad last updated on 18/Mar/18

$i t h i n k t h e Q . i s f i n d {l i m}_{n \to \infty} \sum_{k = 1}^{n} \frac{1}{n} s i n (\frac{2 k π}{n})$ $l e t p u t S_{n} = \sum_{k = 1}^{n} \frac{1}{n} s i n (\frac{2 k π}{n}) w e h a v e$ $S_{n} = \frac{1}{2 π} \frac{2 π}{n} \sum_{k = 1}^{n} s i n (k \frac{2 π}{n}) \to \frac{1}{2 π} \int_{0}^{2 π} s i n x d x$ $= \frac{1}{2 π} {[- c o s x]}_{0}^{2 π} = 0 \Rightarrow {l i m}_{n \to \infty} S_{n} = 0$
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