Re: series example
- From: "rancid moth" <rancidmoth@xxxxxxxxx>
- Date: Tue, 14 Aug 2007 07:03:06 +1000
<C6L1V@xxxxxxx> wrote in message
news:1187034976.024481.100460@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
On Aug 12, 9:16 pm, "rancid moth" <rancidm...@xxxxxxxxx> wrote:
hello,
Problem 11, chapter 2, in WW
What is WW?
R.G. Vickson
whittaker and watson, course of modern analysis fourth edition.
states: in the series whose general term is
u[n]=q^(n-v)*x^(1/2*v*(v+1)) (0<q<1<x) where v denotes the number of
digits
in the expression of n, in the ordinary decimal scale of notation, show
that
lim u[n]^(1/n) =q and that the series is convergent, although lim
u[n+1]/u[n] = oo.
My attempt at a solution is as follows:
v = 1 + [Log(n)] where [] is the floor function and Log is log base 10.
using this in u[n]^(1/n) and noting that lim [Log[n]]/n =0, then (if i
have
everything correct) its shows that lim u[n]^(1/n) =q. The ratio of
terms,
however, prsents an exponent on x having the form
1/2*((1+[Log[n+1])^2-(1-[Log[n]])^2+(1+[Log[n+1]])-(1+[Log[n]]))
the difference (1+[Log[n+1])-(1+[Log[n]]), i think can only ever be 1 or
zero. It will be zero for all numbers n, such that the number of digits
in
n is the same as that in n+1. So for example, it equates to one where
the
number of digits of n+1 and n differs, i.e. at 9, 99, 999, 9999 etc.
The
difference of the squared terms does a similar thing, in that since we
know
(1+[Log[n+1]) is the number of digits of n+1, say N+1, then the
difference
is simply (N+1)^2 - N^2 = 2N+1.
So 1/2*((1+[Log[n+1])^2-(1-[Log[n]])^2+(1+[Log[n+1]])-(1+[Log[n]])) = N+1
where N is the number of digits of n. otherwise, it is zero. The ratio
lim u[n+1]/u[n] as n->oo then approaches oo. But perhaps this is not
entirely correct, since i am tempted to say that the lim u[n+1]/u[n]
doesnt
exist. Or am i incorrect in this assumption? If i am correct in
everything
up to this point, then wouldnt a discussion of its lim inf and lim sup be
appropriate?
I say thins because if i write down that sequence,
{{0^8},2,{0}^88,3,{0}^888,4,....}
where i am using the rather bad notation of {0^n} representing a sequence
of
n zeros, i.e. 0,0,0,0.... I think then that lim inf = 0, since in any
subsequence of terms, zero is going to be included. looking at the
entire
sequence i think the sup is +oo and so, then looking at subsequences i
think
lim sup = +oo, since this is the greatest possible limit of any
subsequence.
Since the lim inf <1, the ratio test in this case, really doesnt tell us
anything about its convergence/divergence, and i think that is what the
question is trying to illustrate.
.
- References:
- series example
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- series example
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