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 ?
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 20220414 10:19:22 