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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) + ...

 Submitted by Federico Kereki Rating: 2.7500 (4 votes) Solution: (Hide) The nth term is n/(k+1+k+2+...+k+n) where k=n(n-1)/2; this works out to n/(nk+n(n+1)/2)= 1/(k+(n+1)/2)= 2/(n²+1).So, we need to calculate the sum for n>0 of 2/(n²+1), which is the same as -1+ the sum for n=-∞ to +∞ of 1/(n²+1).The latter sum equals (proof) πcoth(π), so the final result is πcoth(π)-1.

 Subject Author Date Problem Solution With Explanation K Sengupta 2007-05-28 12:15:57 answer K Sengupta 2007-05-28 12:14:40 re: Solution Sarifa 2004-10-08 17:48:13 re: Solution Nick Hobson 2004-09-01 14:51:37 Solution Keith Steele 2004-08-30 20:10:09 re: Solution Federico Kereki 2004-08-28 21:24:33 Solution Nick Hobson 2004-08-27 16:34:34 solution bernie 2004-08-26 13:53:49 re: starting observation Vee-Liem Veefessional 2004-08-26 00:27:58 re(2): No Subject Vee-Liem Veefessional 2004-08-26 00:23:03 re: No Subject Federico Kereki 2004-08-25 08:51:47 No Subject Vee-Liem Veefessional 2004-08-24 23:36:59 starting observation Charlie 2004-08-24 12:54:58

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