# Thread: deducing a limit...

1. ## deducing a limit...

Deduce that $\displaystyle\sum_{i=1}^n \frac{1}{n+i}\to log 2$ as $n \to \infty$

How on earth do i "deduce this"?! I've tried using $f(x) = \frac{1}{1 + x}$ but that didn't really get me anywhere...

Thanks in advance!

2. It's a Riemann sum:

$\mathop {\lim }\limits_{n \to \infty } \sum\limits_{i = 1}^n {\frac{1}
{{n + i}}} = \mathop {\lim }\limits_{n \to \infty } \frac{1}
{n}\sum\limits_{i = 1}^n {\frac{1}
{{1 + \dfrac{i}
{n}}}} = \int_0^1 {\frac{1}
{{1 + x}}\,dx} .$

The rest follows.