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.

Obviously this is the general result for complex values of 1.

In the special case where 1 takes real values the above expression simplifies to Log(2)/2.

(Just could not resist the joke, sorry!)

Henrik