# Math Help - Log(x+1)

1. ## Log(x+1)

$\int^x_0\sum^\infty_{n=1}\frac{(-1)^{n-1}t^{n}}{n}=\int^x_0\sum^\infty_{n=1}(-1)^{n-1}t^{n-1}dt\underbrace{=}_{\mbox{justify}}\int^x_0\frac{1 }{1+t}dt=\log(1+x)$

Ok so I need to prove that the middle equality it true, but I don't have the slightest clue where to begin.

2. ## Re: Log(x+1)

You have to prove that:
$\sum_{n=1}^{\infty}(-1)^{n-1}t^{n-1}=\frac{1}{1+t}$
It can be useful to write a few terms of the summation:
$\sum_{n=1}^{\infty}(-1)^{n-1}t^{n-1}=1-t+t^2-t^3+...+(-1)^{k-1}t^{k-1}+...$

Do you notice a 'special' serie? Can you continue?

3. ## Re: Log(x+1)

$1-t+t^2-t^3+t^4-....$ is an infinite geometric series with $a=1$ and $r=-t.$

Therefore, $1-t+t^2-t^3+t^4-.... = \frac{a}{1-r}$