Infinite series, definition

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

Convergence of series, definition

Let $\sum _{n=m}^{\mathrm{\infty }}{a}_n$ be a formal infinite series. For any integer $N\ge m$, we define the N^th 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 $\sum _{n=m}^{\mathrm{\infty }}{a}_n$ is convergent, and converges to $L$. We then write

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