Riemann Sums

**acevipa** · June 7th 2010, 02:18 AM

Show that $\lim_{n\rightarrow\infty}\sum_{i=0}^n\frac{1}{n+i} =\log 2$ by writing the sum under the limit as the Riemann sum of an appropriate function.

**tonio** · June 7th 2010, 03:58 AM

Originally Posted by acevipa

Show that $\lim_{n\rightarrow\infty}\sum_{i=0}^n\frac{1}{n+i} =\log 2$ by writing the sum under the limit as the Riemann sum of an appropriate function.

$\lim_{n\rightarrow\infty}\sum_{i=0}^n\frac{1}{n+i} =\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=0}^n\ frac{n}{n+i}=$ $\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=0}^n\f rac{1}{1+i/n}=:\int\limits^2_1\frac{dx}{x}$

Tonio

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