Question

How do you find a power series representation for `f(x)= 1/(1+x)` and what is the radius of convergence?

Answer 1

`sum_(n=0)^oo (-1)^n x^n` with radius of convergence `1`

Explanation:

Start writing out a power series which when multiplied by `(1+x)` gives `1`...

`1 = (1+x)(1-x+x^2-x^3+x^4-...)`

We choose each successive term to cancel out the extraneous term left over by the previous ones.

Then writing it out formally...

`(1+x) sum_(n=0)^N (-1)^n x^n`

`= sum_(n=0)^N (-1)^n x^n + x sum_(n=0)^N (-1)^n x^n`

`= sum_(n=0)^N (-1)^n x^n - sum_(n=1)^(N+1) (-1)^n x^n`

`= (-1)^0x^0 - (-1)^(N+1)x^(N+1) = 1 - (-x)^(N+1)`

So if `abs(x) < 1`, then `(-x)^(N+1) -> 0` as `N->oo` and we find

`(1+x) sum_(n=0)^oo (-1)^n x^n = 1`

Conversely, if `abs(x) >= 1` then `lim_(N->oo) (-x)^(N+1)` does not exist and the sum does not converge.

So the radius of convergence is `1`