If f(x)=∑(n=1 to ∞) (-1)^n*(x-π)^n/n^2 find f '''(x)?

To your first question, yes and no. You start moving up by 1 after you take the first derivative because you see, the term n=1 for f'(x) still exists, it's a scalar. The reason you move it up by 1 in the seceding derivatives is because this (and those of the following derivatives) scalar becomes 0. So n=2 for f''(x) and n=3 for f'''(x)