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
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 [...].