Question

How do I solve this I know the I am supposed to use Alternating Series Test, but I have not encountered (-2)^n, i am used to (-1)^n

http://i.imgur.com/bGGI3yy.png

Thanks

Answer 1

You can just rewrite \((-2)^{n}=2^{n}*(-1)^{n}\)

Answer 2

Do you know the alternating series test?

Answer 3

It has to satisfy that an+1<= an and lim an = 0

Answer 4

in order to be convergent

Answer 5

Rewrite that series in terms of \(a_{n}=(-1)^{n}*b_{n}\). If \(b_{n}\) decreases and \(lim_{n \rightarrow \infty}b_{n}=0\), \(a_{n}\) converges..

Answer 6

Are you having trouble on finding \(b_{n}\)?

Answer 7

is bn just 2^n/7^n

Answer 8

What about the n?

Answer 9

I mean, there's a n multiplying what you just posted.

Answer 10

Oh right my bad so bn=n2^n / 7^n right?

Answer 11

yes. You can just write that as \(n*(\frac{2}{7})^{n}\)

Answer 12

Can you finish it?

Answer 13

I am having troubles finding the limit. can i just use l'hopitals rule?

Answer 14

Yes! You will need l'hospital's rule to do it. But you need to rewrite it first before applying the rule.

Answer 15

Alternatively, you can notice pretty easily that \(7^{n}\) grows WAY faster than \(n*2^{n}\). But as we are doing the "formal" way, just solve the limit.

Answer 16

Ah yes 7^n grows much faster than n*2^n therefore the limit is zero.

Answer 17

Yes. You might try doing the limit though.

Answer 18

Now to finish the alternating series test you need if the sequence { \(b_{n}\) } decreases. do you know how?

Answer 19

I would use the first derivative test on bn. to See if it eventually decreases

Answer 20

You just need to check if \(b_{n}>b_{n+1}\). Your method would work too, but you need to be careful with the intervals!

Answer 21

You can now finish the question without problems.

Answer 22

If i took the absolute value of an |an| = would i get n*2^n/7^n

Answer 23

Yes.

Answer 24

But you just need to check the limit and if the sequence \(b_{n}\) decreases really.

Answer 25

Oh okay. thanks!

Answer 26

Oh, right, we need to check for absolute convergence!

Answer 27

I missed that. But it seems you got it.

Answer 28

So by taking the absolute value it would no longer be an alternating series?

Answer 29

Yes. By taking |an| you check for absolute convergence.