Hi guys,
I'm having real difficulty with this
Show that
, where .
Now, I know by L'Hopital's rule, that
But the relation of n with k has really thrown me off.
Any help would be much appreciated.
Thanks in advance,
HTale.
Your problem amounts to interchanging a limit and an infinite product (or a series), can you see why?
You need to show . Due to your remark, this can be rewritten as
where if and if .
You need a theorem to justify that. A good one would be the "dominated convergence theorem" (you may know another name):Dominated convergence theorem (for series):I wrote the theorem for series, so you must let in order to have and apply the theorem.
If is such that
- for every , ;
- there is such that for every , and ,
Then, for every , the series and converge, and we have .
Thus you must find such that converges and, for every ,
.(And the inequality is obvious if since we would have )
To this aim, you can use the following bounds: if , then . I let you carry on from there.
Note: I assumed so that the factors of the product are positive and the logarithm makes sense.
Hi guys,
I'm having real trouble proving that
as
The following hint was provided,
HINT: choose an m to be much smaller than n, which approaches infinity (for example, or ) and separately consider the part of the product from which r goes from 1 to m and the part from m+1 to n. Taking the logarithm and bounding the sum from r=m+1 to r=n one should be able to prove that the second product is close to 1. As for the first product ranging from 1 to m, one must show that the numerator and denominator of each term has ratio very close to 1.
So, does that mean starting off with something like,
I have no idea how to continue with this.
Thank you all so very very much in advance,
Regards,
HTale.
PS: Just incase it's too small to read, the bracketed part with the sin's read
for the numerator, and
for the denominator.
Ok, I have not made any sense of the hint (it was provided by the author of the book from the exercise I'm trying to solve), but if we take the logarithm of the above, and manipulating it a bit, we get
,
which is the same as saying,
.
Now, here's the interesting bit. By L'Hopitals rule
.
So now I'm appealing to any analysts here; is there some theorem out there which allows me to do
Any help would be much appreciated.
Thanks in advance,
HTale.
First of all, your manipulation result is incorrect: . You dropped the "log"-s in the process.
Then, l'Hopital's rule does not make sense here, since the RHS is also a limit on . Remember: r is an integer between 1 and k ! So the r-s close to k tend to too.
However,
when and . (Because when )
The first condition is satisfied when , cause .
But the second condition is not, cause r goes from 1 to k and when r = k for instance , which is not .
But that's when separating the sum / product comes into play.
if you limit the range of r to 1 to m = integer_part( ), then , you get to have when .
So the 1st part of the sum / product vanishes out smoothly. -> wrong
No ! in fact the 2nd part of the product vanishes (tends to 1).
The first part does not. Even if every r-term tends individually to 1, their product doesn't. (Give me some time to find a counter-example).
The first part tends to 0, because intuitively,
as , there are more and more r-terms with "close" to , which makes the denominator of the r-term close to 0. More and more of these r-terms accumulate, dragging the product close to 0.
I guess I'm being a bit unclear with the explanation ... Can someone help me out ?