Difference between sequence of functions and series

Question

I am currently studying uniform and pointwise convergence and I am stuck at a somewhat basic distinction. Until now our lecturer has been talking about sequences of functions. For example:

$$f_n(x)=\frac{n}{x} \space \space \text{or} \space \space f_n(x)=x^n$$

However, in our problem set I am supposed to test the following series for uniform convergence:

$$\sum_{n=0}^{\infty}\frac{\sin(nt)}{e^n}$$

Are sequences of functions and series related? Is the process of showing uniform convergence different for series?

Answer 1

From any series $\sum_{k=0}^{\infty} a_k$, you can construct the sequence of partial sums, $$ S_n = \sum_{k=0}^n a_k, $$ and treat it as you would a sequence. Conversely, given any sequence $(b_n)_{n=0}^{\infty}$, the differences $$ a_0 = b_0, \quad a_k = b_k-b_{k-1} $$ can be made into a series, $$ b_n = \sum_{k=0}^{n} a_k. $$ This is much like differentiation and integration being "inverse operations" (scare quotes required for technical reasons, of course).

Answer 2

You can view a series as a sequence of partial sums. Try to think of the sequence of functions $$f_i(t)=\sum_{n=0}^i \frac{\sin(nt)}{e^n} $$