# Expansion of log(1+x)

1. Sep 17, 2011

We know that log(1+x) = x+((x^2)/2)+((x^3)/3)+.............((x^n)/n)+...........
Could any body please tell me the proof

2. Sep 17, 2011

### micromass

Staff Emeritus
3. Sep 17, 2011

### mathman

log(1+x)=x - x2/2 + x3/3 - x4/4 + ....
(Alternate signs)

The easiest way to see it is by using an integral representation.

log(1+x) = ∫dx/(1+x)

Since 1/(1+x) = 1 - x + x2 - x3 + .....,

integrating term by term gives the series for log(1+x), where the integration limits are [0,x].

4. Sep 17, 2011

thankyou for the answers i am grateful to you

5. Sep 17, 2011

In taylor series if f(x)=log(1+x) then is f'(a)=0?
and is f(a)=log(1+a)

Last edited: Sep 17, 2011
6. Sep 18, 2011

### gb7nash

Assuming a > -1:

What do you get when you take the derivative of log(1+x)? What do you get when you plug a in?