# Math Help - Elementary Analysis

1. ## 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

2. 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.

3. 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?

4. Yes it is 1b.

5. 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.

6. 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).

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

8. 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.

9. 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?

10. Originally Posted by Jhevon