Understanding series and their sums Here's something that I can't wrap my head around while self-studying analysis. Is defining a function to be a series and defining a function to be the sum of a series considered to be two different concepts, or do they mean the same thing? 
For example, if I defined $f$ as $$f(x) = \sum_{n=0}^\infty a_nx^n$$ and asked about the differentiability of $f$, would that signify the same question as defining $g$ to be the sum of above series (that is, the function it converges to), and asking if $g$ is differentiable?
What triggered this question was the following theorem:
https://proofwiki.org/wiki/Power_Series_Differentiable_on_Interval_of_Convergence
Again, as asked above, would this theorem give us that $f$, the sum, is differentiable, or that the power series itself is differentiable and that this is something else entirely?
 A: Something appears to be wrong with your notation. a sum of a series would be the sum of two series. where a series is defined to be of the form
\begin{equation}
\sum_{n=x}^{y}a_n
\end{equation}
where x and y are natural numbers and a_n is defined to be a sequence. A sequence is defined to be a mapping from the natural numbers onto the real line. A power series is an infinite series of the form
\begin{equation}
\sum_{n=0}^{\infty}a_nx^n
\end{equation}
So a power series is an infinite polynomial (pretty cool if you ask me). When we say that it is differentiable, it means to say that what the series converges to is differentiable. So, now we should think about what it means for an infinite sum to converge. The reality of the matter is that we can't actually take an infinite sum, but rather show that for large sums we can get as close as we want to a particular value. One can read on the Cauchy criterion to get a better understanding of this. So when we say that the infinite sum is differentiable, we really mean that what it converges to (if anything) is differentiable. However, one can use the Cauchy criterion and further prove that the derivative of the thing it converges to, is indeed the "derivative of the sum". That is to say that
\begin{equation}
f'(x)=\sum_{n=1}^{\infty}na_nx^{n-1}
\end{equation}
Real analysis is a very rich and beautiful study. It often requires that we prove things about the infinite by examining the finite. I hope this may have helped you better understand some of the most important concepts in this abstract field of mathematics.
