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   Author  Topic: A sequence  (Read 728 times)
wmat1234
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A sequence  
« on: Aug 31st, 2007, 7:18am »		 Quote Quote Modify Modify
Sequence:
 
x_1 = 1
 
x_n = x_{n-1} + sqrt(x_{n-1})
 
what is lim x_n/n^2
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Eigenray
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Re: A sequence  
« Reply #1 on: Aug 31st, 2007, 9:21am »		 Quote Quote Modify Modify
Let yn = 2xn; then
 
yn+12 = yn2 + 2yn.
 
On the one hand,
 
yn+12 < (yn+1)2,
 
so yn < n, and limsup yn/n 1.
 
On the other hand,
 
Clearly xn xn-1 + 1, so xn (hence yn) goes to infinity.  Now, fix 0<a<1.  Then there exists an N such that for n>N, yn > C = a2/[2(1-a)], and therefore
 
yn+12 = yn2 + 2yn > yn2 + 2ayn + a2 = (yn + a)2,
 
so for n>N, yn > C + a(n-N).
 
Therefore liminf yn/n a.  Since this holds for all a<1, we have liminf yn/n 1.  Since also yn < n, it follows lim yn/n = 1, and thus lim xn/n2 = 1/4.
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