How to Show Log(1+x) = x-x^{2}\2 + x^{3}\3 ........ using power series?

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- Jan 20th 2013, 06:41 AMsshPower series solution for Log(1+x)
How to Show Log(1+x) = x-x

^{2}\2 + x^{3}\3 ........ using power series? - Jan 20th 2013, 12:47 PMHallsofIvyRe: Power series solution for Log(1+x)
I don't understand what you mean by "using power series". What you give

**is**a power series.

There are a number of different ways you can show that power series sums to log(1+x). The simplest, probably, is to show that it has a property that you know only log(1+ x) has.

That is, that the derivative of log(1+ x) is $\displaystyle \frac{1}{1+ x}$ and that log(1+ 0)= log(1)= 0. (There are an infinite number of functions satisfying f'(x)= 1/(1+ x) but only one of them has f(0)= 0.)

So, differentiate that series "term by term". The result is, of course, another power series. So find the "geometric series" (if the problem had said "using**geometric**series", it would have been pointing directly to this method) that sums to 1/(1+ x)= 1/(1- (-x)) and compare.

Oh, and if x= 0, every term of that power series is 0.