April 4th, 2017, 06:46 AM  #11 
Math Team Joined: Dec 2013 From: Colombia Posts: 6,685 Thanks: 2175 Math Focus: Mainly analysis and algebra 
[QUOTE=zylo;566136All the natural numbers appear as suffixes in both sums [/QUOTE] So they are two series, not one. Their sum is defined only if both series converge separately. I agree that there is something counterintuitive about $1+3+5+\ldots +2+4+6+\ldots$ not being a rearrangement of $1+2+3+\ldots$. But from the point of view that $1+3+5+\ldots$ is a sum that never ends, we can clearly never get to the even numbers. 

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rearrangement, series 
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