it's a little ugly, because you have to have some strong conditions to use associativity on infinite series (and I forget what they are off the top of my head). Of course, this is true for splitting up the e^x into cos(x) and sin(x) as well.
The series has to be absolutely convergent. That is, you can rearrange the terms of \sum_{i=0}^\infty a_n freely if and only if \sum_{i=0}^\infty |a_n| converges. See http://en.wikipedia.org/wiki/Riemann_series_theorem
Comments
Here is another way (a bit informal):
e = lim_{n->infinity} (1 + 1/n)^n
Now, apply the binomial theorem:
1 + n * 1/n + n! / (2 (n-2)! n^2) + ... + n! / (m! (n - m!) n^m) + ...
Now, for each m, we have this sequence:
a_n = n! / (m! (n - m)! n^m)
Which converges on 1/m!, so we are left with this:
1 + 1 + 1/2! + 1/3! + 1/4! + ...
it's a little ugly, because you have to have some strong conditions to use associativity on infinite series (and I forget what they are off the top of my head). Of course, this is true for splitting up the e^x into cos(x) and sin(x) as well.
The series has to be absolutely convergent. That is, you can rearrange the terms of \sum_{i=0}^\infty a_n freely if and only if \sum_{i=0}^\infty |a_n| converges. See http://en.wikipedia.org/wiki/Riemann_series_theorem