What is the purpose of differentiating or integrating an infinite series? I'm reading "A Radical Approach to Real Analysis" by Bressoud, which I'm enjoying a lot. He is explaining the problem of dealing with infinite cosine series that Fourier had. Then he starts rambling on about integrating/differentiating the series (regarding the heat transfer problem). 
I understand integrating/ differentiating a function will keep the radius of convergence intact, but could increase or decrease the convergence interval.  
My question is why worry about integrating/differentiating a series at all? 
 A: Fourier was interested in solving the heat equation $({\partial\over \partial t}-\Delta_x)u=0$, a differential equation. He wanted to "separate variables", writing a "Fourier series" in the spatial variable $x$, with coefficients depending on "time"  $t$. The basic utility of this is that cosines and sines (or, equivalently, exponentials $x\to e^{inx}$) are eigenfunctions for the Laplacian $\Delta$. But the whole computation is stuck if we cannot apply $\Delta$ term-by-term...
A: The short answer is: 

We integrate and differentiate series of functions for the same reason why we want to integrate and differentiate a finite sum of functions (e.g. polynomials, trigonometric functions) - it is because they are functions. It is natural for us to investigate differentiation (local properties) and integration (global properties) of any given function. 

The problem of studying infinite series as opposed to ordinary functions of finite terms is that our basic operations (addition and multiplication) are defined only for finite terms. Therefore, when it comes sum/product of infinitely many terms, there would be ambiguity since after all, our basic operations only work finitely.
One way to work around the ambiguity of infinity is to introduce the concept of convergence. With the notion of convergence (induced by topologies), we could define like the case for finite sum what an infinite sum of function means. Nonetheless, this simply solve the problem of definitions. To deal with practical purpose, we want to investigate properties of infinite series by simply looking at the finite sum, or by its terms.
What you meant in your question by changing the radius of convergence, or the convergence interval, of a series of functions upon differentiation and integration exactly illustrates some problems when we try to approach infinite sum with its finite terms. It is not always possible to investigate properties of an infinite series (whose definition requires more than basic algebraic operations) with only its finite terms. One basic example is the limit and summation interchange condition. Placing a limit into a summation allows you to do about the finite terms before summing infinitely; placing out is the other way round. 
