Series Expansion: Proving ln(1+x)/x

[thercias]

Homework Statement

Show that 1- x/2 + x^2/3 - x^3/4 + x^4/5... (-1)^n (x^n)/(n+1) = ln(1+x)/x
with |x| < 1

Homework Equations

The Attempt at a Solution

finding derivative of the function multiplied by x
d/dx(xS(x))
= 1 -x + x^2 - x^3 + x^4 - x^5 +...

absolute value of this function = 1 + x + x^2 + x^3 + x^4... = 1/1-x ( geometric series)
the integral of that gives us -ln(1-x) + c

drawing a blank now. not really sure where to go.

 

[haruspex]

= 1 -x + x^2 - x^3 + x^4 - x^5 +...

absolute value of this function = 1 + x + x^2 + x^3 + x^4... = 1/1-x ( geometric series)

No it isn't. But it is a geometric series, so you can sum it quite easily.

 

[thercias]

ok the absolute value part is wrong, but what do you mean i can sum it up easily? i don't get it. is the integral still -ln(1-x) + c..? i know the geometric series for 1 + x + x^2 + ... = 1/1-x, but how does that change with alternating signs? is it going to be (-1)^n*(1/1-x)? so the integral is (-1)^n*-ln(1-x)? how do i go from that to ln(1+x)/x?

 

[HallsofIvy]

An alternating series involves (1)ⁿ which can easily be incorported into the geometric series.

 

[haruspex]

An alternating series involves (1)ⁿ which can easily be incorported into the geometric series.

I think you meant (-1)ⁿ.
Thercias, what is the ratio of consecutive terms in your alternating series? If it's independent of n then it's a geometric series.