Can someone explain to me why: Given that 1/(1-x) is represented by the power series 1 + x + x^2 + ... + x^n + ... on the interval (-1,1) the power series that represents (1/(1+x)) = 1 - x + x^2 - x^3 + ... (-x)^n on (-1,1). I'm trying to wrap my head around power series, but I don't quite understand.

Question

Can someone explain to me why: Given that  1/(1-x) is represented by the power series 1 + x + x^2 + ... + x^n + ... on the interval (-1,1)  the power series that represents (1/(1+x)) = 1 - x + x^2 - x^3 + ... (-x)^n on (-1,1). I'm trying to wrap my head around power series, but I don't quite understand.

anonymous · Answer

there is binomial theorem to prove this....you can check....
but here i can tell you to prove it using GP series....
1+x+x^2.......n terms...=(x^n-1)/(x-1)
now if |x|<1 and n->infinity so x^n->0. so you get
1+x+x^2+....x^n+....=1/(1-x)
and hence your result...I think you can follw it it is easier to understand...