Give a closed expression for the infinite sum:
1/1 + 2/(2+3) + 3/(4+5+6) + 4/(7+8+9+10) + ...
Consider the nth term.
The last term in its denominator is the nth triangular number: ½n(n+1).
So the first term in its denominator is ½n(n-1) + 1.
The mean of these two terms is ½(n²+1).
As there are n terms in arithmetical progression in the denominator, the nth term equals n/[½n(n²+1)] = 2/(n²+1).
The sum from n = 1 to infinity of 2/(n²+1) may be evaluated using contour integration.
The answer is pi×coth(pi) - 1, where coth(x) = (ex + e-x)/(ex - e-x).
(It strikes me this last step is quite difficult for a level 3, so I suspect Federico may have another solution up his sleeve!)
Solution
