Question

Alternating series Q:

Answer 1

In the book there is an example that's \[\sum_{n=1}^{\infty} \frac{n}{(-2)^{n-1}}\]

Answer 2

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...

Answer 3

Slight note: Monotonic over [1, infty]

Answer 4

how or why? the how is clear right? if \(n>1\) then \(\frac{n+1}{n}>\frac{1}{2}\) for sure

Answer 5

Well that's a property of your book's proof. If you like, we can probably do an ALTERNATE one (get it?).

Answer 6

NVM. The point is, whatever your book is doing to prove the alternating series converges, it needs that fact...

Answer 7

Well do you have to make a formal proof? Or an AP Calc BC-suitable proof?

Answer 8

Do we have to do stuff like this, or just apply the rules in the theorm to see if it wortks or not?

Answer 9

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?

Answer 10

The course I'm in is Calc 2.

Answer 11

The book only says 2 conditons an+1 < an and an = 0.

Answer 12

I did the question I was goign to do and was correct, but I did do it a diff way than the professor..

Answer 13

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.

Answer 14

Seems like a lot of L'hospitals rule is used.. Should bone up.

Answer 15

oh yeah it says let an > 0

Answer 16

LH is a good thing to always keep in your mind.

Answer 17

so hmm... can you check out the first picture and see what they are doing?

Answer 18

not sure if it's important for my needs.