term by term differentiation of $\sum_{n \geq 1} \arctan \frac {x} {n^2}$
I was reading a book and found a question "analyze the applicability of term by term differentiation for $\sum_{n \geq 1} \arctan \frac {x} {n^2}$", What does that means? How to solve this?
Solution 1:
You are being asked to investigate whether or not $$\frac{d}{dx}\sum_{n\ge1}\arctan\left(\frac{x}{n^2}\right)=\sum_{n\ge1}\frac{d}{dx}\arctan\left(\frac{x}{n^2}\right)$$ I believe that answers your first question. Your second question is how to accomplish this. But now that the first is more clear, you might have some ideas.