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Question Number 108474 by pticantor last updated on 17/Aug/20

$${\mathit{U}}_{\mathit{n}}=\underset{k=0}{\sum ^{2n-1}}\frac{1}{2n+k}=?$$ $$\text{Double subscripts: use braces to clarify}$$

Answered by Dwaipayan Shikari last updated on 17/Aug/20

$$\underset{k=0}{\sum ^{2n-1}}\frac{1}{n(2+\frac{k}{n})}=\frac{1}{n}\underset{n\to \mathrm{\infty }}{\lim }\underset{k=1}{\sum ^{2n}}\frac{1}{2+\frac{k}{n}}={\int }_0^2\frac{1}{2+x}dx=log{\left[(x+2)\right]}_0^2=log\left(2\right)$$

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