Question 1037098
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What do you want to do with it?

I'll do what I can:

SUM = 1 - 4/5 - 3/5 - 8/17 - 5/13 - ...????

SUM = 2/2 - 4/5 - 6/10 - 8/17 - 10/26 - ...????

The sequence of absolute value of numerators is 2,4,6,8,10
The sequence of absolute value of denominators is 2,5,10,17,26

2,4,6,8,10 has general term 2n
1,5,10,17,26 has general term n<sup>2</sup>+1

The general term would be {{{-2n/(n^2+1)}}} except for the first
term being positive instead of negative, so we write the first term as
{{{2-2/2}}}

SUM = 2 - 2/2 - 4/5 - 6/10 - 8/17 - 10/26 - ...????

So it could be written this way:

SUM = {{{2 - sum((2k/(k^2+1)),k=1,n)}}}

The next term after -10/26 (or -5/13) would be {{{-2(6)/(6^2+1)=-12/(36+1)=-12/37}}}

It can't be summed to infinity because it diverges when compared 
to the harmonic p-series <font face="symbol">S</font>(1/n).

Edwin</pre>