Ponder This Challenge - July 2001 - 1/1 + 2/(2+3) + 3/(4+5+6) + ...

Find an exact expression for the infinite sum:

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

By "exact expression", I mean that "sqrt(2)" would be accepted
(if that were the right answer) but "1.414..." would not.

(This problem was posed by Saradakanta Ganguli, a long time ago.)

Solution

• Click here to view the solution

  Here we let i=sqrt(-1) be the imaginary unit; pi=3.14...;

  and a**b denotes exponentiation.

  The nth term is n/(k+1+k+2+...+k+n) where k=n(n-1)/2.
  This is n/(nk+n(n+1)/2) = 1/(k+(n+1)/2) = 2/(n**2+1).
  So we want sum(2/(n**2+1)) as n ranges from 1 to infinity.

  This is the same as
  i(-1/i + sum(1/(n+i))),
  where in this sum n ranges from -infinity to infinity.

  In turn, sum(1/(n+x)) (from -infinity to infinity) is
  (pi)*cotangent(pi*x), so that our sum becomes

  i(i+(pi)*cotangent(pi*i))
  = -1 + pi*coth(pi) (the hyperbolic cotangent)
  = -1 + pi*( e**pi + e**(-pi) ) / ( e**pi - e**(-pi) )

  ---

  If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com

Solvers

• Henry Bottomley (07.02.2001 @ 12:37 PM EDT)
• Vince Lynch (07.02.2001 @ 12:57 PM EDT)
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