Math and science::Analysis::Tao::07. Series

# Infinite series, definition

Tao breaks the definition into two pieces (reasons discussed on flip side). I present the 2nd part first. See the flip side for the initial definition of a formal series.

### Convergence of series, definition

Let $$\sum_{n=m}^{\infty} a_n$$ be a formal infinite series. For any integer $$N \ge m$$, we define the Nth partial sum $$S_N$$ of this series to be

[...]

$$S_N$$ is thus a real number.

If the sequence [...] converges to some limit $$L$$, then we say that the infinite series $$\sum_{n=m}^{\infty} a_n$$ is convergent, and converges to $$L$$. We then write

$L = \sum_{n=m}^{\infty} a_n$

and say that $$L$$ is the sum of the infinite series $$\sum_{n=m}^{\infty} a_n$$. If the sequence of partial sums $$(S_N)_{N=m}^{\infty}$$ diverge, then we say that the infinite series $$\sum_{n=m}^{\infty} a_n$$ is divergent, and we [...].