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Summing up II (Posted on 2022-04-14) Difficulty: 3 of 5
S = 1 + 1/(1+2) + 1/(1+2+3) + … + 1/(1+2+3+4+…+n)

Find a simple expression for S

What is the limit of S when n ==> infinity ?

See The Solution Submitted by Ady TZIDON    
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Solution Solution Comment 2 of 2 |
The denominators of S are the triangular numbers which are given by n(n+1)/2.  Hence S(n)=2/(n(n+1)).

Observe this fraction can be decomposed into S(n) = 2(1/n - 1/(n+1))

So if you sum the terms from 1 to n, most of these fractions cancel and all that remains is 

2(1 - 1/(n+1)) = 2 - 2/(n+1)

Which tends to 2 as n-->infinity

  Posted by Jer on 2022-04-14 10:19:22
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