# Strange NSum behavior

If I do:

 NSum[(i + 1)/(i + 2) LegendreP[i, 0] LegendreP[i, 0], {i, 0, Infinity}]


I get:

1.25216


If I do:

Sum[(i + 1)/(i + 2) LegendreP[i, 0] LegendreP[i, 0], {i, 0, 1000}] // N


The result is:

2.40862


This last result is larger than the infinity sum, but the sum terms are all nonnegative. That is not possible. Am I wrong or is Mathematica?

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The odd terms are zero. –  Giovanni Aug 28 '13 at 17:29
One thing is that I am not sure if the sum diverges. If this is the case, the infinity sum is wrong anyway. –  Giovanni Aug 28 '13 at 17:53
The sum diverges. –  Giovanni Aug 28 '13 at 18:14
But see here SumConvergence[(i + 1)/(i + 2) LegendreP[i, 0] LegendreP[i, 0], i, Assumptions -> Element[i, Integers] && Mod[i, 2] == 0]! –  PlatoManiac Aug 28 '13 at 18:17
Interesting. I did an Integral test and it diverged. –  Giovanni Aug 28 '13 at 18:38