Need some slight help on the following sum, this is my first post so i hope ive put this in the correct place
sum of from k=2 to infinity (1/(k^3-k)
currently i have the following, ive turned it to a partial fraction and obtained
sum of from k=2 to infinity (k/(k^2-1)-(1/k)
Then from this entering numerical values for k:
(2/3-1/2+3/8-1/3......) and so on however cant see any relation between numbers cancelling etc like in previous questions i have done.
Thanks in advance for any help.
Trying to go down both roots here, i think the one that im after is via the partial fraction but struggling. However going for the general rule and i obtained 1/(k/4(k(k^2+2k+1)-2(k+1)) not sure if this is correct from here to rearrange to get the formula stated. I used the sum of k up to infinity being k/2(k+1) then subtracted the sum up to 1, the same for k^3 using k^2/4(k+1)^2?
i split that up into partial fractions getting the sum of from 2 to infinity 1/2(k-1)-1/2(k+1) which turns out i think to be 1/2(1+1/2)=3/4 then we have the subtraction of k=2 to infinity for 1/k, i decided to put it as one sum and try to spot of patterns etc as i have done previous hence sum of k=2 to infinity of
1/2( 1/(k-1)-1/(k+1)-2/k) to get the series of 1/2(1-1/3-1+1/2-1/4/-2/3+1/3-1/5-1/2....) a lot of the terms cancel but cant see anything of such so not sure what to do....