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