let give U_n =n ( (1/n^2 ) + (1/(1^2 +n^2 ))+ (1/(2^2 +n^2 )) +.... (1/((n−1)^2 +n^2 ))) find lim_(n−>∝) U


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Question Number 27663 by abdo imad last updated on 12/Jan/18

$l e t g i v e U_{n} = n (\frac{1}{n^{2}} + \frac{1}{1^{2} + n^{2}} + \frac{1}{2^{2} + n^{2}} + . . . . \frac{1}{{(n - 1)}^{2} + n^{2}})$ $f i n d {l i m}_{n - >\propto} U_{n} .$
Commented byabdo imad last updated on 13/Jan/18

$w e h a v e U_{n} = n \sum_{k = 0}^{n - 1} \frac{1}{k^{2} + n^{2}} = \frac{1}{n} \sum_{k = 0}^{n - 1} \frac{1}{1 + {(\frac{k}{n})}^{2}}$ $U_{n} i s a R i e m a n s u m a n d$ ${l i m}_{n - >\propto} U_{n} = {l i m}_{n - >\propto} \frac{1 - 0}{n} \sum_{k = 0}^{n - 1} \frac{1}{1 + {(\frac{k (1 - 0)}{n})}^{2}}$ $= \int_{0}^{1} \frac{d x}{1 + x^{2}}^{} = {[a r c t a n x]}_{0}^{1} = \frac{π}{4} .$
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