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

##### 1 Answer

Oct 5, 2015

#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 = (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

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

Conversely, if

So the radius of convergence is