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.
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
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.
Define a sequence of non-zero terms as,(b) lim(1/n)(n!)^(1/n)
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).