SOLUTION: Σn=1 2/(n(n+1)) ITS AN INFINITE SERIES AND STARTS AT 1.

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Question 1133418: Σn=1 2/(n(n+1))
ITS AN INFINITE SERIES AND STARTS AT 1.
Answer by ikleyn(52788) About Me  (Show Source):
You can put this solution on YOUR website!
.

The requested sum is  (it is how I read the problem)


2%2F%281%2A2%29 + 2%2F%282%2A3%29 + 2%2F%283%2A4%29 + . . . 2%2F%28n%2A%28n%2B1%29%29 + . . . to infinity = 


        Each term  2%2F%28n%2A%28n%2B1%29%29 = 2%2Fn - 2%2F%28n%2B1%29.



        Therefore, any finite sum   2%2F%281%2A2%29 + 2%2F%282%2A3%29 + 2%2F%283%2A4%29 + . . . 2%2F%28n%2A%28n%2B1%29%29 = 


        = (2%2F1 - 2%2F2) + (2%2F2 - 2%2F3) + (2%2F3 - 3%2F4) + . . . + (2%2Fn - 2%2F%28n%2B1%29) = all interior terms cancel each other, 


           and only two extreme (very first and very last) terms survive


         = 2 - 2%2F%28n%2B1%29.



It implies that the limit of the sum  at  n -->oo  is  2 (two, TWO).   ANSWER.

Solved.