lim_(n→∞) Σ_(r=1) ^∞ (r/(n^2 +r))


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Question Number 174630 by infinityaction last updated on 06/Aug/22

$\lim_{n \to \infty} \underset{r = 1}{\sum^{\infty}} \frac{r}{n^{2} + r}$
	Answered by mnjuly1970 last updated on 06/Aug/22
	$lim_(n→∞) {(1/(1+n^( 2) )) +(2/(2 +n^( 2) )) +(3/(3+n^( 2) )) +...(n/(n+n^( 2) )) =a_( n) } ≤ (1/(1+n^( 2) )) +(2/(1+n^( 2) )) +...+(n/(1+n^( 2) )) = ((n(n+1))/(2(1+n^( 2) ))) (1) a_n ≥(1/(n+n^( 2) )) +(2/(n+n^( 2) )) +...+(n/(n+n^( 2) )) = ((n(n+1))/(2n(1+n))) (1/2) ≤ a_( n) ≤ ((n(n+1))/(2(1+n^2 ))) lim_(n→) (a_( n) )= (1/2)$
	${l i m}_{n \to \infty} {\frac{1}{1 + n^{2}} + \frac{2}{2 + n^{2}} + \frac{3}{3 + n^{2}} + . . . \frac{n}{n + n^{2}} = a_{n}}$ $⩽ \frac{1}{1 + n^{2}} + \frac{2}{1 + n^{2}} + . . . + \frac{n}{1 + n^{2}}$ $= \frac{n (n + 1)}{2 (1 + n^{2})} (1)$ $a_{n} ⩾ \frac{1}{n + n^{2}} + \frac{2}{n + n^{2}} + . . . + \frac{n}{n + n^{2}}$ $= \frac{n (n + 1)}{2 n (1 + n)}$ $\frac{1}{2} ⩽ a_{n} ⩽ \frac{n (n + 1)}{2 (1 + n^{2})}$ ${l i m}_{n \to} (a_{n}) = \frac{1}{2}$
	Commented by infinityaction last updated on 06/Aug/22

	$t h a n k s s i r$
	Commented by mnjuly1970 last updated on 06/Aug/22

	$y o u a r e w e l c o m e s i r . . .$
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