I have the function:$$f(x)=\sum_{n=1}^{\infty}\frac{ (-1)^{n+1} }{ 2n }x^n, x \in]-R,R[$$ And I need to find the Maclaurin series for $$f´(x)$$I have found the convergence radius to 1. Now since R0, then I can use term-by-tern differentiation right?

Question

I have the function:$$f(x)=\sum_{n=1}^{\infty}\frac{ (-1)^{n+1} }{ 2n }x^n,   x \in]-R,R[$$ And I need to find the Maclaurin series for $$f´(x)$$I have found the convergence radius to 1. Now since R>0, then I can use term-by-tern differentiation right?

anonymous · Answer

Why do we need $R>0$?

anonymous · Answer

I wanted to use term-by-term differentiation. And that can only be done if we have a power series with radius of convergence $$R>0$$

anonymous · Answer

Right, you can use term-by-term differentiation.

anonymous · Answer

I am not 100% how to use that. Would that give me $$f'(x)=\sum_{n=2}^{\infty}\frac{ (n+1)(-1)^n }{ 2 }x^n$$ @mukushla

anonymous · Answer

$(x^n)'=n x^{n-1}$ and you don't need to change the lower bound