Question Number 196614 by mr W last updated on 28/Aug/23
$${if}\:{S}_{{n}} =\frac{\mathrm{1}}{\mathrm{1}+\mathrm{5}{n}}+\frac{\mathrm{1}}{\mathrm{2}+\mathrm{5}{n}}+\frac{\mathrm{1}}{\mathrm{3}+\mathrm{5}{n}}+…+\frac{\mathrm{1}}{\mathrm{6}{n}}, \\ $$$${find}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}{S}_{{n}} =? \\ $$
Answered by universe last updated on 28/Aug/23
$$\mathrm{log}\:\mathrm{6}/\mathrm{5} \\ $$
Commented by mr W last updated on 28/Aug/23
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Answered by sniper237 last updated on 28/Aug/23
$${S}_{{n}} =\:\frac{\mathrm{1}}{{n}}\:\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{\mathrm{5}+\frac{{k}}{{n}}}\:\underset{\infty} {\rightarrow}\:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{dx}}{\mathrm{5}+{x}}={ln}\left(\frac{\mathrm{6}}{\mathrm{5}}\right) \\ $$
Commented by mr W last updated on 28/Aug/23
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