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 \( (S_N)_{N=m}^{\infty} \) 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 do not assign any real value to the series.
Formal infinite series, definition
A formal infinite series is any expression of the form
where \( m \) is an integer and \( a_n \) is a real number for any integer \( n \ge m \).
Tao mentions that at this point, this series is only defined formally; in other words, the series hasn't been set to equal anything. [This is all fine, I just don't see why we needed this initial step].
Necessary?
My take is that we could just use a definition like so:
With this definition, the ideas of the infinite sum only taking values when the limit converges is inherited from the definition of the limit. Tao's presentation seems to repeat a lot of these ideas again.
Update on my take: so far at least, limits are only defined in terms of sequences. Tao is explicit about what sequence the limit is operating over. My approach does not make this explicit and is assuming that Tao's step of defining the corresponding sequence occurs.
Tao even continues his use of the term formal series further into the chapter and I haven't seen a clear transition at which the terminology is defined away.