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 Nice sum (Posted on 2004-08-24)
Give a closed expression for the infinite sum:

1/1 + 2/(2+3) + 3/(4+5+6) + 4/(7+8+9+10) + ...

 See The Solution Submitted by Federico Kereki Rating: 2.7500 (4 votes)

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 Solution | Comment 7 of 13 |

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!)

 Posted by Nick Hobson on 2004-08-27 16:34:34

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