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Question Number 153172 by SANOGO last updated on 05/Sep/21
lim_ _(x→+oo) Σ_(k=o) ^n^2  (n/( n^2 +k^2 ))
$$\underset{{x}\rightarrow+{oo}} {\mathrm{li}\underset{} {{m}}}\underset{{k}={o}} {\overset{{n}^{\mathrm{2}} } {\sum}}\frac{{n}}{\:{n}^{\mathrm{2}} +{k}^{\mathrm{2}} } \\ $$
Answered by Ar Brandon last updated on 05/Sep/21
L=lim_(n→∞) Σ_(k=0) ^n^2  (n/(n^2 +k^2 ))=lim_(n=∞) (1/n)Σ_(k=0) ^n^2  (1/(1+(k^2 /n^2 )))       =lim_(n→∞) ∫_0 ^n (dx/(1+x^2 ))=lim_(n→∞) [arctan(x)]_0 ^n =±(π/2)
$$\mathscr{L}=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\underset{{k}=\mathrm{0}} {\overset{{n}^{\mathrm{2}} } {\sum}}\frac{{n}}{{n}^{\mathrm{2}} +{k}^{\mathrm{2}} }=\underset{{n}=\infty} {\mathrm{lim}}\frac{\mathrm{1}}{{n}}\underset{{k}=\mathrm{0}} {\overset{{n}^{\mathrm{2}} } {\sum}}\frac{\mathrm{1}}{\mathrm{1}+\frac{{k}^{\mathrm{2}} }{{n}^{\mathrm{2}} }} \\ $$$$\:\:\:\:\:=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\int_{\mathrm{0}} ^{{n}} \frac{{dx}}{\mathrm{1}+{x}^{\mathrm{2}} }=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left[\mathrm{arctan}\left({x}\right)\right]_{\mathrm{0}} ^{{n}} =\pm\frac{\pi}{\mathrm{2}} \\ $$
Commented by SANOGO last updated on 05/Sep/21
stp explique moi sur les bornes la de o a  n
$${stp}\:{explique}\:{moi}\:{sur}\:{les}\:{bornes}\:{la}\:{de}\:{o}\:{a}\:\:{n} \\ $$
Commented by Ar Brandon last updated on 06/Sep/21
lim_(n→∞) Σ_(k=1) ^(tn) f(x_k )=∫_0 ^t f(x)dx
$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\underset{{k}=\mathrm{1}} {\overset{{tn}} {\sum}}{f}\left({x}_{{k}} \right)=\int_{\mathrm{0}} ^{{t}} {f}\left({x}\right){dx} \\ $$

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