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Homework Help: A summation question

  1. May 8, 2006 #1
    HELP: A summation question


    Given the sum

    [tex]\sum _{p=0} ^{\infty} (-1)^p \frac{4p+1}{4^p}[/tex]

    I have tried something please tell if I'm on the right track

    Looking at the alternating series test

    (a) [tex]1/(4^{p+1}) < (1/(4^p))[/tex]

    (b) [tex]\mathop {\lim }\limits_{p \to \infty } b_p = \mathop {\lim }\limits_{p \to \infty } \frac{1}{{4^p }} = 0[/tex]

    Then according to the test this allows me to write [tex]\sum _{p = 0} ^{\infty} 4^{-p} = 4/3[/tex]

    Can anybody please verify if I'm heading in the right direction on this? Or am I totally wrong?

    Sincerely Yours

    Last edited: May 8, 2006
  2. jcsd
  3. May 8, 2006 #2
    I agree that you've shown this series converges, but I don't see where you're getting that you can say it converges to [itex]\frac{4}{3}[/itex]. The Alternating Series Test can show conditional convergence, but not a numerical value to the best of my knowledge.
  4. May 8, 2006 #3
    Okay thanks I can see that now,

    but what would be the next logical step to find the sum of this series? Should I use a specific test?

    Sincerely Yours

    p.s. Since it converges, then |1/(4^p)| < 1 ??

    Last edited: May 8, 2006
  5. May 8, 2006 #4


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    Homework Helper

    This is another of those that can be broken up with one of the summands being of the form [tex]px^p[/tex]. Remember the method I suggested in your other thread ?
  6. May 9, 2006 #5
    Hello and the other sum being

    (-1)^p ?

    Sincerely Hummingbird25

  7. May 9, 2006 #6


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    Homework Helper

    No, the other summand is [tex](-4)^{-p}[/tex].

    That's just a geometric series.