Sorry, but I can't understand what you mean. What is the question exactly?
And what does this mean? What have you multiplied by (n+1) and where?so the correct is what I have multiplied by (n+1)
Represent the function f(x) = x/(1+4x)^2 as a power series and determine the radius of convergence.
so the correct answer is (-1)^n * (n+1) *4^n*x^(n+1) but this is what I got:
x/(1+4x)^2
x/[1-(-4x)^2
x series sign (-4x)^n
series sign (-1)^n * 4^n x^(n+1)
Where did the (n+1) in front of the 4^n come from?
You showed in the OP that you already know the series for 1/(1 - x). The derivative of that series is the series of that derivative, -1/(1 - x)^2. This more closely resembles the original problem which you are working towards.
This is just one tool you can use to find a series for a function you didn't know.