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 n2+1

The general term would be  except for the first
term being positive instead of negative, so we write the first term as


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

So it could be written this way:

SUM = 

The next term after -10/26 (or -5/13) would be 

It can't be summed to infinity because it diverges when compared 
to the harmonic p-series S(1/n).

Edwin