# Riemann Sums

• June 7th 2010, 02:18 AM
acevipa
Riemann Sums
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.
• June 7th 2010, 03:58 AM
tonio
Quote:

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