Evaluation of a divergent series?

[standardflop]

Hello,
according to theory of alternating series, the series \sum_{n=0}^\infty (-1)^n is not convergent, correct?. Howcome maple estimates it as \sum_{n=0}^\infty (-1)^n = 0.5000000000. This seems strange to me.. Why this strange result?

 

[HallsofIvy]

Maple clearly has a bug in it! I suspect it has calculated for very large n, calculated for n+ 1 and since they were different, averaged the two answers!

 

[shmoe]

It's not a bug, but related to how Maple handles infinite sums. From the description of Maple's sum command:

"Note that sum knows about various resummation methods and will thus be able to give the 'correct' value for various classes of divergent sums. If one wants to restrict summation to convergent sums, then explicit convergence checks must be done."

I'm not sure what method it's using in this case, but it agrees with Cesaro summation, where you essentially take the average of the partial sums. Even though your series is Cesaro summable, it's still divergent in the 'usual' sense (though maple doesn't report this).