A -very important- POV on series:
- synthetic
  formal series
- analytic
  do they represent functions?
NB: I'm not a huge fan of the words; this dichotomy is overused.

For polynomials,
- synthetic: (formal) power series
- analytic: Taylor series

For trig functions,
- synthetic: (formal) trig series
- analytic: Fourier series

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This is important b/c it causes much confusion.
- the -original motivation- for studying series is to represent
  functions by them

- formal series are -very easy to work with-;
  that is their attraction

- summing
  the sum of a series is generally the limit of its partial sums,
  though there are other possible definitions (Cesaro, etc.);
  as function spaces admit several topologies, it's important to ask
  -in which topology- you mean this.
  [EG, for Taylor series you want the compact-open,
   though people often just say the pointwise.]

  It's always obvious that finite sums converge; duh.

- expansions
  a way to go from functions to series is called an "expansion"
  for Taylor series/Fourier series,
  it's obvious that the series of (the function represented by
  a finite sum) is the original sum;
    (which shows this is reasonable)
  the interest is in what happens for infinite series; duh.

- understanding which formal operations are still legit at the level
  of functions is much subtler (EG, swapping sums and integrals)

- issues of convergence are often very delicate
  [it's quite complicated for Fourier series,
   and even for power series (where the picture is easy),
   the results are wacky (EG, for composition/inversion/reciprocal of
   series)]
