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let-S-n-k-1-n-1-n-2-2kn-find-lim-n-S-n-




Question Number 31509 by abdo imad last updated on 09/Mar/18
let S_n =Σ_(k=1) ^n    (1/( (√(n^2  +2kn))))  find  lim_(n→∞)  S_n .
$${let}\:{S}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\:\frac{\mathrm{1}}{\:\sqrt{{n}^{\mathrm{2}} \:+\mathrm{2}{kn}}}\:\:{find}\:\:{lim}_{{n}\rightarrow\infty} \:{S}_{{n}} . \\ $$
Commented by abdo imad last updated on 13/Mar/18
we have S_n =(1/n)Σ_(k=1) ^n    (1/( (√(1+((2k)/n))))) =(1/2) (2/n)Σ_(k=1) ^n   (1/( (√(1+((k(2−0))/n)))))  ⇒lim_(n→∞)  S_n   =(1/2) ∫_0 ^(2   )    (dx/( (√(1+x))))=∫_0 ^2   (dx/(2(√(1+x))))=[(√(1+x)) ]_0 ^2   =(√3) −1 .
$${we}\:{have}\:{S}_{{n}} =\frac{\mathrm{1}}{{n}}\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+\frac{\mathrm{2}{k}}{{n}}}}\:=\frac{\mathrm{1}}{\mathrm{2}}\:\frac{\mathrm{2}}{{n}}\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+\frac{{k}\left(\mathrm{2}−\mathrm{0}\right)}{{n}}}} \\ $$$$\Rightarrow{lim}_{{n}\rightarrow\infty} \:{S}_{{n}} \:\:=\frac{\mathrm{1}}{\mathrm{2}}\:\int_{\mathrm{0}} ^{\mathrm{2}\:\:\:} \:\:\:\frac{{dx}}{\:\sqrt{\mathrm{1}+{x}}}=\int_{\mathrm{0}} ^{\mathrm{2}} \:\:\frac{{dx}}{\mathrm{2}\sqrt{\mathrm{1}+{x}}}=\left[\sqrt{\mathrm{1}+{x}}\:\right]_{\mathrm{0}} ^{\mathrm{2}} \\ $$$$=\sqrt{\mathrm{3}}\:−\mathrm{1}\:. \\ $$

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