Convergence/Divergence and Reordering of an Alternating Series

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SUMMARY

The series 1 - 2 + 3 - 4 + 5... is divergent, as established in the discussion. Reordering the series to -1 + 2 - 3 + 4 - 5... does not change its divergence; it can be rearranged to approach both positive and negative infinity. The series does not converge to any particular value, demonstrating the Riemann series theorem, which states that the sum of a conditionally convergent series can be altered by rearrangement.

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I was just thinking about the following series:

1-2+3-4+5...

I'm not familiar with any other series like this one(other than the alternating harmonic), and I was curious as to whether or not it would be convergent, and if reordering it to

-1+2-3+4-5...

would change its convergence.

If its divergent, which it seems it would be, is it divergent to negative or positive infinity?

This is probably a basic question, but I'm unable to figure it out right now. Any takers?
 
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It is divergent to no particular value. It can be made to approach anything by suitable rearrangement. Examples:
-1+2 = 1, -3+4=1, etc. -> +∞.
0-1=-1, 2-3=-1, 4-5=-1, etc. -> -∞.
 

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