The original poster was asking about beating this information out of Wolfram|Alpha.
I got part way by splitting the problem. First I did the sum,
sum (i/(i^2 + n^2)), i=1 to n
I then used the Copyable PlainText link to get the output into another W|A window. I wrote the Limit out fully.
Limit as n goes to infinity of 1/2 (-polygamma(0, 1-i n)-polygamma(0, i n+1)+polygamma(0, (1-i) n+1)+polygamma(0, (1+i) n+1))
That tried but ran out of time.
simplify (-polygamma(0, 1-i n)-polygamma(0, i n+1)+polygamma(0, (1-i) n+1)+polygamma(0, (1+i) n+1))
Gave a sum of Harmonic numbers, which did not look nicer to deal with.