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Homework Help: Basic Geometric Series Question

  1. Aug 23, 2012 #1
    1. The problem statement, all variables and given/known data
    Determine whether the series is convergent or divergent. Find the sum if possible

    Ʃ 1+2^n / 3^n n=1 -> infinity

    2. Relevant equations

    3. The attempt at a solution

    I split it up so that the equation is now:

    Ʃ (1/3^n) + (2/3)^n n=1 -> infinity
    Ʃ (1/3^n) + Ʃ (2/3)^n n=1 -> infinity

    I know that it is convergent in the second term because 2/3 < 1, how do I setup a/1-r for term 1? :S

    a1=1/3 a1 = 2/3
    (1/3)/(1-(1/3)) + (2/3)/(1-(2/3))

    The answer should be 3/2 on the answer sheet it says ?
  2. jcsd
  3. Aug 23, 2012 #2


    Staff: Mentor


    I don't see anything wrong with your answer, but I had a hard time trying to figure out what your problem was.

    This is what you wrote (fixed up in LaTeX):
    $$ \sum_{n = 1}^{\infty} \left( 1 + \frac{2^n}{3^n}\right)$$

    This is what I'm pretty sure you meant:
    $$ \sum_{n = 1}^{\infty} \left(\frac{1 + 2^n}{3^n}\right)$$

    Don't write 1+2^n / 3^n if you mean (1+2^n )/ 3^n.
  4. Aug 23, 2012 #3
    The first sum, once you've broken them up, is not a geometric sum. Think about what that guy's doing for a little bit.
  5. Aug 23, 2012 #4


    Staff: Mentor

    Sure it is.
    The first series can be written as
    $$\sum_{n = 1}^{\infty}\left(\frac{1}{3}\right)^n $$

    or 1/3 + (1/3)2 + (1/3)3 + ... +
  6. Aug 23, 2012 #5
    derp. right.
  7. Aug 24, 2012 #6


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    Had I been Mark44, I would have issued you a warning or infraction for that last sentence.
  8. Aug 24, 2012 #7
    I apologize. Personification is against the rules?
    Last edited: Aug 24, 2012
  9. Aug 24, 2012 #8


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    I'm not the boss around here but I just wanted to alert you that your comment might easily be construed as violating the section below. It struck me that way anyway.

    From the Rules menu at the top of the page:

    Language and Attitude: Foul or hostile language will not be tolerated on Physics Forums. This includes profanity, obscenity, or obvious indecent language; direct personal attacks or insults; snide remarks or phrases that appear to be an attempt to "put down" another member; and other indirect attacks on a member's character or motives.
  10. Aug 24, 2012 #9
    Ah. When I said "that guy," I was referring to the first sum in the OP's question. I guess I see how that could be misconstrued.
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