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Question Number 203059 by hassanmpsy last updated on 08/Jan/24

Commented by witcher3 last updated on 11/Jan/24

$${\mathrm{U}}_{\mathrm{n}}=\underset{\mathrm{k}=1}{\sum ^{\mathrm{n}}}\frac{\mathrm{n}(1+\frac{\mathrm{k}}{\mathrm{n}})}{{\mathrm{n}}^2(2+2\frac{\mathrm{k}}{\mathrm{n}}+{\left(\frac{\mathrm{k}}{\mathrm{n}}\right)}^2)}=\frac{1}{\mathrm{n}}\underset{\mathrm{k}=1}{\sum ^{\mathrm{n}}}\mathrm{f}\left(\frac{\mathrm{k}}{\mathrm{n}}\right)$$$$\text{Double subscripts: use braces to clarify}$$$$=\frac{1}{2}{\left[\ln ({\mathrm{x}}^2+2\mathrm{x}+2)\right]}_0^1=\frac{1}{2}\ln \left(3\right)$$

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