Could anyone help me correcting my formula? I tried to enter the following limit of a sum in Wolfram alpha, but it doesn't seem to understand it correctly.
limit (sum (i/(i^2 + n^2)), i=1..n) as n->infinity
Any suggestions are very welcome!
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.
Appreciate the joke. I should have applied FullSimplify before posting (or done a bit more thinking).
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!)
Using Mathematica 10.0.1 the limit is
which is approximately 0.346574.