# Thread: infinite series problem with log

1. ## infinite series problem with log

This problem is giving me a hard time, i have no clue how to solve for it to be able to state if its convergent or divergent. The problem is:
signma : log(n+1) - log(n) and the limits are n=1 to infinity

thanks in advance for the help

2. Originally Posted by [s]arah

This problem is giving me a hard time, i have no clue how to solve for it to be able to state if its convergent or divergent. The problem is:
signma : log(n+1) - log(n) and the limits are n=1 to infinity

thanks in advance for the help
$\displaystyle \sum_{i=1}^\infty \ln (n+1) - \ln n$

Stop the series at N and write out some of the terms

$\displaystyle S_N=\sum_{i=1}^N\ln (n+1) - \ln n =$ $\displaystyle \left(\ln 2 - \ln 1\right) + \left(\ln 3 - \ln 2\right) + \left(\ln 4 - \ln 3\right) + \cdots + \left(\ln N - \ln (N-1)\right) + \left(\ln (N+1) - \ln N\right)$

then cancel terms giving

$\displaystyle S_N = \sum_{i=1}^N\ln (n+1) - \ln n = \ln (N+1)$

Then $\displaystyle \lim_{N \to \infty } S_N$