Re: find the limit of this quotient

On Feb 8, 11:47 pm, magidin@xxxxxxxxxxxxxxxxx (Arturo Magidin) wrote:
In article <36986358-f991-493f-91d2-328a91c84824@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Randy Poe <poespam-trap@xxxxxxxxx> wrote:

On Feb 8, 11:14 pm, "Carl R." <solrac...@xxxxxxxxxxx> wrote:
Let a_n be a sequence of real numbers which converges to 0.

Question: What are the possible limits which can take the quotient
which consists of the sequence
a_(n+1)/a_n when n tends to infinity?

Days ago, Ken Pledger showed the limit is 1 when the sequence
converges to a nonzero limit.
But somewhere in the proof he used the fact that the limit is
nonzero to define epsilon as |l|/4 * epsilon.
But what happens when the original sequences converges to 0?

First, I think we need two things: to prove the sequence converges
and then find the value of the limit.
How can you do this?
I think the answer is 1 and is the only possible value of the limit
of the quotient but how do you prove this?

Counterexample:

a_n = 1/n! -> 0

a_(n+1)/a_n = n!/(n+1)! = 1/(n+1) -> 0

It all depends on how fast a_n converges to 0.

If a_n converges to 0, then the ratio must converge to a
value <= 1.

In fact, of absolute value less than or equal to 1.

(If the absolute value is greater than 1, then |a_{n+1}|>|a_n| for all
sufficiently large n, hence the sequence cannot converge to 0).

But I think that the limit could be anything in [0,1]. In
fact I suspect it could be anything in [-1,1].

You've exhibited one with limit 0; to get one with limit 1, take
a_n = 1/n. To get one with limit -1, take a_n = (-1)^n/n. Then
a_{n+1}/a_n = (-1)^{n+1}*n/(-1)^n*(n+1) = -n/(n+1), with limit -1.

And for any nonzero r, -1 < r < 1, let a_n = r^n. Then lim a_n = 0,
and lim(a_{n+1}/a_n) = lim(r^{n+1}/r^n) = lim(r) = r.

--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================

Arturo Magidin
magidin-at-member-ams-org

this is interesting. But just wondering if you can generalize the
idea. As in can you always prove that if the sequence converges to
zero (lim a_n=0), there will be a subsequence a_n_k such that a_n_(k
+1)/a_n_k=0?
.

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