Thursday, 10 March 2016

Sum of n terms of the series frac11cdot3+frac21cdot3cdot5+frac31cdot3cdot5cdot7+cdots




I need to find the sum of n terms of the series



113+2135+31357+



And I've no idea how to move on. It doesn't look like an arithmetic progression or a geometric progression. As far as I can tell it's not telescoping. What do I do?


Answer



It is telescoping. Consider that:
113=12(1113),2135=12(113135),
31357=12(11351357),

so:
nk=1k(2k+1)!!=12(11(2n+1)!!).
As usual, (2k+1)!! stands for 135(2k+1).


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