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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

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  1. ChillOut
    • one year ago
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    You can just rewrite \((-2)^{n}=2^{n}*(-1)^{n}\)

  2. ChillOut
    • one year ago
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    Do you know the alternating series test?

  3. anonymous
    • one year ago
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    It has to satisfy that an+1<= an and lim an = 0

  4. anonymous
    • one year ago
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    in order to be convergent

  5. ChillOut
    • one year ago
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    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
    • one year ago
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    Are you having trouble on finding \(b_{n}\)?

  7. anonymous
    • one year ago
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    is bn just 2^n/7^n

  8. ChillOut
    • one year ago
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    What about the n?

  9. ChillOut
    • one year ago
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    I mean, there's a n multiplying what you just posted.

  10. anonymous
    • one year ago
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    Oh right my bad so bn=n2^n / 7^n right?

  11. ChillOut
    • one year ago
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    yes. You can just write that as \(n*(\frac{2}{7})^{n}\)

  12. ChillOut
    • one year ago
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    Can you finish it?

  13. anonymous
    • one year ago
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    I am having troubles finding the limit. can i just use l'hopitals rule?

  14. ChillOut
    • one year ago
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    Yes! You will need l'hospital's rule to do it. But you need to rewrite it first before applying the rule.

  15. ChillOut
    • one year ago
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    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
    • one year ago
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    Ah yes 7^n grows much faster than n*2^n therefore the limit is zero.

  17. ChillOut
    • one year ago
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    Yes. You might try doing the limit though.

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

  19. anonymous
    • one year ago
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    I would use the first derivative test on bn. to See if it eventually decreases

  20. ChillOut
    • one year ago
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    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
    • one year ago
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    You can now finish the question without problems.

  22. anonymous
    • one year ago
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    If i took the absolute value of an |an| = would i get n*2^n/7^n

  23. ChillOut
    • one year ago
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    Yes.

  24. ChillOut
    • one year ago
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    But you just need to check the limit and if the sequence \(b_{n}\) decreases really.

  25. anonymous
    • one year ago
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    Oh okay. thanks!

  26. ChillOut
    • one year ago
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    Oh, right, we need to check for absolute convergence!

  27. ChillOut
    • one year ago
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    I missed that. But it seems you got it.

  28. anonymous
    • one year ago
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    So by taking the absolute value it would no longer be an alternating series?

  29. ChillOut
    • one year ago
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    Yes. By taking |an| you check for absolute convergence.

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