## KonradZuse 3 years ago Alternating series Q:

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

2. vf321

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...

3. vf321

Slight note: Monotonic over [1, infty]

4. satellite73

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

5. vf321

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

6. vf321

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

7. vf321

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

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

10. vf321

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?

The course I'm in is Calc 2.

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

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

14. vf321

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.

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

oh yeah it says let an > 0

17. vf321

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

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