Series Order of Summation For certain series, if you change the order of the terms, the series will converge to a different number. Where can I find the proof for this and could someone explain why this does not contradict the fact that addition is commutative? 
 A: If a series $\sum a_n$ converges absolutely, then you can rearrange the terms in any manner, and the sum will converges to the same number.  If $\sum a_n$ is conditionally convergent, you can rearrange the terms to converge to any limit you like.  You can find these results in any standard text in Analysis.  Rudin's Principles of Mathematical Analysis or Apostol's Mathematical Analysis are good sources. 
As for why this doesn't contradict the commutativity of addition, the commutative law says
$a+b=b+a$ for any $a,b\in\Bbb R$. Here each sum consists of two terms.  This can be generalized to any number of finite terms by a standard induction argument.  However, it does not extend to infinite sums.
A: The result you speak of goes by the name "Riemann's Series Theorem".
It's not contradictory to $+$ being commutative, because essentially addition is a binary operation, which can be extended to be an $n$-ary operation for any $n$ by induction. The theorem thus says the commutativity of addition is preserved for countable entries if (and only if) the series is absolutely convergent.
A: This is known as Riemann's rearrangement theorem. You can find a proof in Spivak's Calculus (ch. 22, theorem 6 of my edition) or on Wikipedia.
