Math Help Forum: Elementary Analysis

So i got 8 questions for homework, i did 6. Here are the last two i'm having problems with. Thanks a lot guys.

12.12
Let (Sn) be a sequence of nonnegative numbers, and for each n, define
An = (1/n)(s1 + s2 + ... + sn)

(a) Show that liminfSn <= liminf An <= limsupAn <= limsupSn

Hint: for the last inequality show first that M > N implies sup{An : n>M} <= (1/M)(s1 + s2 + ... + sN) + sup{Sn : n>N}

(b) Show that if limSn exists, then limAn exists and limAn = limSn

12.14
Calculate:
(a) lim(n!)^(1/n)
(b) lim(1/n)(n!)^(1/n)

So yea, that last problem seems like a calc 3 problem, but nevertheless, i dont know how to do it.

Another thing that i probably should know already: Is adding sequences like adding vectors? as in you add corresponding terms of the sequences? What about multiplying two sequences, how is that done?

I may come back and post my solutions for the questions i did just to see what you guys think, but only if i can find the time, this homework is due monday. thanks again

Without TeX this next to impossible. Above I have done one case for you.
Now let the sequence (S_n – L) converges to zero. Use that to finish.

Originally Posted by Plato

Without TeX this next to impossible. Above I have done one case for you.
Now let the sequence (S_n – L) converges to zero. Use that to finish.

this is for 1(b) right?

Yes it is 1b.

Originally Posted by Jhevon

Calculate:
(a) lim(n!)^(1/n)

Ah! These were the fun ones for homework.

Let s_n=n!

Note, that this is always non-zero.
By the ratio test,
lim |(n+1)!/n!|=lim (n+1) = +infinity.

That means that, the root test always gives the same result.
Hence,
lim |n!|^{1/n}=+\infinity.

(b) lim(1/n)(n!)^(1/n)

Define a sequence of non-zero terms as,
s_n=n!/(n^n)

Then, by the ratio test, (details omitted)
lim (n/(n+1))^n=1/lim(1+1/n)^n = 1/e.

That means, the root test also gives the same result.
Thus,
lim |s_n|^(1/n)=1/n*(n!)^(1/n)=1/e

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I did the proof Plato did differently from the hint in the book and differently from him.
Since, LaTeX is disabled I cannot post it. Perhaps tomorrow I can show it you.

Here is my proof of that problem.


Note the inequalities #1 and #2 used in proof.
#1 is trivial but #2 takes some work to show.

I also use the fact that,
(a+b)/(c+d)<= a/c+b/d for b,d>0

I do not know if that makes sense to you.
(EDIT. I just realized in the last line I forgot to write out the full sum. But I need to leave now cannot fix it, hopefully you understand what I am talking about).

TPH. That is a nice proof.
But, is it not the same as I gave?

Originally Posted by Plato

TPH. That is a nice proof.

I would not say that it is no nice. Because it is a bit too long (especially when inequality #2 is proven).

But, is it not the same as I gave?

It has some resembelance. But my proof is not based on yours, rather that I how I written it.

Originally Posted by Jhevon

Another thing that i probably should know already: Is adding sequences like adding vectors? as in you add corresponding terms of the sequences? What about multiplying two sequences, how is that done?

What about these questions?

Originally Posted by Jhevon

What about these questions?

Yes.

The book implicity assumes that.

But if {s_n} is a sequence and {t_n} is a sequence.
(And I would imagine have the same domain. But it is standard for sequences to have the entire N domain).

Then, {s_n+t_n} is a sequence {u_n} defined to
u_n=s_n+t_n for all n.