KonradZuse
  • KonradZuse
Alternating series Q:
Mathematics
chestercat
  • chestercat
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KonradZuse
  • KonradZuse
In the book there is an example that's \[\sum_{n=1}^{\infty} \frac{n}{(-2)^{n-1}}\]
anonymous
  • anonymous
Well if n->infinity, then n/(n+1) is 1 since n goes to infinity at the same rate. We also know that the base case, n = 1, IS 1/2, and that the function's monotonic, so n/(n+1) HAS to be between 1/2 and 1...
anonymous
  • anonymous
Slight note: Monotonic over [1, infty]

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anonymous
  • anonymous
how or why? the how is clear right? if \(n>1\) then \(\frac{n+1}{n}>\frac{1}{2}\) for sure
anonymous
  • anonymous
Well that's a property of your book's proof. If you like, we can probably do an ALTERNATE one (get it?).
anonymous
  • anonymous
NVM. The point is, whatever your book is doing to prove the alternating series converges, it needs that fact...
anonymous
  • anonymous
Well do you have to make a formal proof? Or an AP Calc BC-suitable proof?
KonradZuse
  • KonradZuse
KonradZuse
  • KonradZuse
Do we have to do stuff like this, or just apply the rules in the theorm to see if it wortks or not?
anonymous
  • anonymous
lol it was ap calc bc. Nice guess on my part, huh? Yeah for AP you just say "this converges by ALT series test" because of the three conditions for alternating series. Do you know them?
KonradZuse
  • KonradZuse
The course I'm in is Calc 2.
KonradZuse
  • KonradZuse
The book only says 2 conditons an+1 < an and an = 0.
KonradZuse
  • KonradZuse
I did the question I was goign to do and was correct, but I did do it a diff way than the professor..
anonymous
  • anonymous
Well in that case I can't guess your course needs... and the third condition is that the "non alternating part" has to be always positive.
KonradZuse
  • KonradZuse
Seems like a lot of L'hospitals rule is used.. Should bone up.
KonradZuse
  • KonradZuse
oh yeah it says let an > 0
anonymous
  • anonymous
LH is a good thing to always keep in your mind.
KonradZuse
  • KonradZuse
so hmm... can you check out the first picture and see what they are doing?
KonradZuse
  • KonradZuse
not sure if it's important for my needs.

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