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| Limiting Ratio (Posted on 2011-04-22) |
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F(n) is a function defined as:
F(n) = Σj=1 to n j6/2j
Evaluate this limit:
Limit F(n)
n → ∞
Analytic solution
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Comment 3 of 3 | 
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F(n) = Sum(j=1 to n) of j6(1/2)j
Define: f(x) = Sum(j=1 to infinity) of j6xj
f(x) = 16x + 26x2 + 36x3 + 46x4 + ...
Then: lim F(n) as n -> infinity = f(1/2)
f(x) can be constructed from the geometric series: x + x2 + x3 +.... = x/(1-x)
by repeating the following two steps six times: differentiate wrt x then multiply
by x. Thus, using D(y) to denote dy/dx:
f(x) = x*D(x*D(x*D(x*D(x*D(x*D(x/(1-x)))))))
Working out the RHS is tedious, but with the help of partial fractions and a few recurrence tricks you get:
f(x) = x(1 + 57x + 302x2 + 302x3 + 57x4 + x5)/(1 - x)7
Thus: Required limit = f(1/2) = 9366.
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Posted by Harry
on 2011-04-23 00:04:46 |
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