Question

Calculus help

Answer 1

Recall that the Alternating Series Test has three conditions associated with it:
1. The series must alternate.
2. The terms must decrease (in absolute value) for large n.
3. The nth term must go to 0.

Can you even have a series where the first two conditions hold and the third doesn't?
Create a convergent series that satisfies conditions 1 and 3, but not 2.
Create a divergent series that satisfies conditions 1 and 3, but not 2.

Answer 2

does the nth term go to zero?

Answer 3

The first question's scenario can definitely happen e.g. \(\sum\limits_{n=0}^\infty (-1)^n \frac{n}{n+1}\)

Answer 4

yep thx

Answer 5

Sorry, I fell asleep.

Answer 6

@oldrin.bataku don't the terms of your series increase as n increases?

So it wouldn't hold for the 2nd condition for the question "Can you even have a series where the first two conditions hold and the third doesn't?" n/(n+1) approaches 1 (from below) as n gets larger.

Answer 7

so what should be the answer

Answer 8

@agent0smith yeah I typed that from my phone, I think I meant $$\frac{n+1}n$$

Answer 9

Ah yes, that does get smaller as n grows.

Answer 10

what about last two questions

Answer 11

Idk, I didn't spend a lot of time on it, but didn't think of anything for the last two questions.

because i'm wondering how to get a series in which the nth term goes to zero, but the terms don't decrease, for satisying condition 3 but not 2:

2. The terms must decrease (in absolute value) for large n.
3. The nth term must go to 0.