Google Groups no longer supports new Usenet posts or subscriptions. Historical content remains viewable.
Dismiss

Proof/limit help?

0 views
Skip to first unread message

rhh8

unread,
Dec 1, 2009, 10:22:07 AM12/1/09
to
I have to find the limit of the infinite product of (1+(b-1)/(n+k(b-1)). I've tried about a zillion methods (multiplying by 1/n, factoring out, factoring in, common denominators...) and I keep getting the wrong answer (the correct answer is b). Could any of you give me tips or hints please?

Arturo Magidin

unread,
Dec 1, 2009, 11:36:01 AM12/1/09
to
On Dec 1, 9:22 am, rhh8 <theinformant...@yahoo.com> wrote:
> I have to find the limit of the infinite product of (1+(b-1)/(n+k(b-1)).

Your parentheses are unbalanced; you have four left parentheses, but
only three right ones. I would guess you did *not* mean

(1 + (b-1))/(n+k(b-1))

as that would just be b/(n+k(b-1)), simpler to write. So you may have
meant something else. What did you mean, exactly?

And by "the limit", do you mean

lim{n-->oo} Prod_{i=1 to n} (whatever the correct term is)

?

--
Arturo Magidin

rhh8

unread,
Dec 1, 2009, 4:23:56 PM12/1/09
to
sorry, the one is outside the parenthesis I guess:

1+(b-1)/
( k+n(b-1))

I hope that shows up correctly this time.

rhh8

unread,
Dec 1, 2009, 4:21:23 PM12/1/09
to
I'm sorry, I should have been more clear. b>1, and I'm finding the limit as n...infinity, and the product as k...n. Thanks very much for your help anyway, Don!

eratosthenes

unread,
Dec 2, 2009, 6:16:54 AM12/2/09
to
OK. So we need to take the limit of 1+(b-1)/(k+n*(b-1)) with b>1 as
n goes to infinity.

Assuming that this is in fact the correct the correct expression then
the limit is trivial. The denominator in the second term grows toward
infinity, causing the entire second term to shrink towards zero,
causing the entire expression to go towards one. Cheers.

Patrick.

Don Coppersmith

unread,
Dec 2, 2009, 1:28:37 PM12/2/09
to

Write out a few of the factors (k=1, k=2, k=3...)
and see whether you can beat it into the form of a
"telescoping product" (analogous to a "telescoping sum"),
something that looks like
(4/3)*(5/4)*(6/5)*(7/6)
so that you can cancel numerator from one term and
denominator from another.


Also: it is helpful to post the entire problem in one post.
As it is, the reader has to gather bits and pieces from
three different posts, and then try to decide which one
is the correct form, n+k(b-1) or k+n(b-1).

Don Coppersmith

eratosthenes

unread,
Dec 2, 2009, 1:48:50 PM12/2/09
to

Oops. I missed the part about it being an infinite product and not
just a limit. Sorry 'bout that.

Patrick

0 new messages