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Question

# Let Sn,n=1,2,3...., be sum of infinite geometric series whose first term is n and the common ratio is 1n+1. Evaluatelimn→∞S1Sn+S2Sn−1+S3Sn−2+....+SnS1S21+S22+....+S2n.

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Solution

## As we know∫baf(x)dx=limh→0f(a+rh)More generally ∫k0f(x)dx=limh→01nkn∑r=1f(rn) So, from given question And we also know the formula of sum of infinte G.P which is a1+r (where a and r used from usual notation)so writing given question in general form limn→∞∑nr=1(Sn−r+1)(Sr)∑nr=1{Sr2} whereSr=(r)(r+1)(r+2)= limn→∞∑nr=1{(n−r+1)(n−r+2)n−r+3}{(r)(r+1)(r+2)}∑nr=1{(r)(r+1)(r+2)2}Then we divide and multiply by n5,and splitting each n into each bracket for making the given expression as limit as sum and we write rn=x and as tending to infinty 1n=0 so after that we got ∫10{(1−x)(1−x)(1−x)}∫10{(x)(x)(x)}{∫10(x)(x)(x)}2=∫10{1−x}{x}{∫10x}2 and after simple integration we got 12×12(12)2=1

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