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Sum to Infinity - 1 (Posted on 2006-03-11) Difficulty: 4 of 5
Find the sum of the following series:

1 + 4/7 + 9/49 + 16/343 + .......... to infinity

No Solution Yet Submitted by Ravi Raja    
Rating: 3.7143 (7 votes)

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re(2): solution --- seems to be wrong | Comment 3 of 12 |
(In reply to re: solution --- seems to be wrong by Charlie)

The correct answer, I believe, is 392/216=49/27=1.814814814... . I base this on the formula
Sum_{n=1..infinity} n^2*x^n=x*(x+1)/(1-x)^3
for the generating function of n^2 taken from Weisstein's online article "Generating Function." This result can be proved by differentiating the well-known series
1/(1-x)=1+x+x^2+x^3+... twice and doing some arithmetic. These series all converge absolutely for |x|<1, so the differentiation is legitimate. The sum sought is thus 7 times the generating function for n^2, evaluated at x=1/7.

  Posted by Richard on 2006-03-11 13:03:38

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