I need help with this. Find a power series representation for f(x)=x/(1+4x)^2. Find radius of convergence. Thanks.

Question

I differentiated but I got stuck. 

Answer 1

Try taking out the x and integrating.

 

(-x/4) d/dx (1+4x)^-1 = x(1+4x)^-2

 

Then on the left we have

 

(-x/4) d/dx (1 - (-4x))^-1

 

= (-x/4) d/dx ∑ (-4)ⁿxⁿ    by the formula for a geometric series, where the series goes from n=0 to infinity

 

= (-x/4) ∑ n(-4)ⁿx^n-1       by differentiating, where the series goes from n=1 to infinity

= ∑ n(-4)^n-1xⁿ

 

The radius of convergence is when |r| < 1, so when |-4x| < 1

4|x| < 1 

|x| < 1/4