Question

How do you find a power series converging to `f(x)=x/(1+x)^4` and determine the radius of convergence?

Answer 1

`x/(1+x)^4=1/6 sum_(nu=0)^oo(-1)^(nu+1) nu(nu+1)(nu+2)x^nu` for `absx < 1`

Explanation:

There are infinite realizations, each depending to the point in which is done. We will develop a realization for the set `abs x < 1` centered at `x=0`

We know that

`d^3/(dx^3)(1/(1+x)) = -6/(1+x)^4`

and also that

`1/(1+x) = sum_(k=0)^oo (-1)^k x^k` for `abs x < 1` then

`x/(1+x)^4 = -x/6 d^3/(dx^3)(1/(1+x)) = -1/6sum_(k=0)^oo(-1)^k k(k-1)(k-2)x^(k-2)`

and making `nu=k-3`

`x/(1+x)^4 =-x/6 sum_(nu=0)^oo (-1)^(nu+3)(nu+1)(nu+2)(nu+3)x^nu =`

`=1/6 sum_(nu=0)^oo(-1)^(nu+1) nu(nu+1)(nu+2)x^nu`