## anonymous one year ago 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

1. ChillOut

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

2. ChillOut

Do you know the alternating series test?

3. anonymous

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

4. anonymous

in order to be convergent

5. ChillOut

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..

6. ChillOut

Are you having trouble on finding $$b_{n}$$?

7. anonymous

is bn just 2^n/7^n

8. ChillOut

9. ChillOut

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

10. anonymous

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

11. ChillOut

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

12. ChillOut

Can you finish it?

13. anonymous

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

14. ChillOut

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

15. ChillOut

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.

16. anonymous

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

17. ChillOut

Yes. You might try doing the limit though.

18. ChillOut

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

19. anonymous

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

20. ChillOut

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!

21. ChillOut

You can now finish the question without problems.

22. anonymous

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

23. ChillOut

Yes.

24. ChillOut

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

25. anonymous

Oh okay. thanks!

26. ChillOut

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

27. ChillOut

I missed that. But it seems you got it.

28. anonymous

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

29. ChillOut

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