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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 \( \sum_{n=m}^{\infty} a_n \) 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 \( \sum_{n=m}^{\infty} a_n \) 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 [...].