Question

Find a power series representation for the function f(x)= 2/(1-2x). For what values of x does it equal the function?

Answer 1

You can take the Taylor Series Expansion, but I am not so sure what you mean by for what values of x does it equal the function.
In theory all values of x should equal the function so long as the series converge.

Answer 2

Taylor Expansion at center x=0 would be
2+4 x+8 x^2+16 x^3+32 x^4+64 x^5+..

Answer 3

Hmm. Well the geometric progression a+ax^2+ax^3... converges to a/(1-x) when -1<x<1. So let x=ax. Then a+a(ax^2)+a(ax^3)... converges to a/(1-ax) for (-1/a)<x<(1/a). So i guess 2/(1-2x) would be 2+4x^2+8x^3... for (-1/2)<x<(1/2).

Answer 4

Hmm...i think i get what you mean

Answer 5

Should be a+ax+a(ax)^2+a(ax)^3... Missed the x term all over lol

Answer 6

Sigh. Well the geometric progression a+ax+ax^2+ax^3... converges to a/(1-x) when -1<x<1. So let x=ax. Then a+a(ax)+a(ax)^2+a(ax)^3... converges to a/(1-ax) for (-1/a)<x<(1/a). So i guess 2/(1-2x) would be 2+4x+8x^2+16x^3... for (-1/2)<x<(1/2).

Answer 7

so how did you come up w/ it converging to a/(1-x)

Answer 8

Its a geometric series. It's derived without calculus. It goes:
Let a=1+x+x^2+x^3...
So ax=x+x^2+x^3...
a-ax=1.
So a=1/(1-x)

I said it only converges for -1<x<1 because of the ratio test

Answer 9

ok thanks so much! :)

Answer 10

Np