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Homework Help: Radius of convergence

  1. Apr 8, 2007 #1
    1. The radius of convergence of the power series the sum n=1 to infinity of (3x+4)^n / n is

    a 0
    b 1/3
    c 2/3
    d 3/4
    e 4/3

    2. the sum n=1 to infinity of (3x+4)^n / n

    3. no idea

    do the ration test to get abs value 3x+4 < 1 ?
  2. jcsd
  3. Apr 8, 2007 #2
    The ratio test is indeed the way to go. It's always a good place to start when the nth term of the series involves n! or a constant to the nth power.
  4. Apr 8, 2007 #3
    i know but what do i do to get from the ratio test to the radius of convergence?
  5. Apr 8, 2007 #4
    The ratio test says that [itex]\sum_{n=1}^\infty a_n[/itex] is abs. convergent if

    [tex]\lim_{n \rightarrow \infty} \frac{|a_{n+1}|}{|a_n|} < 1,[/tex]

    and divergent if the limit is greater than 1 (assuming in both cases that the limit exists, of course).

    So find out for what [itex]x[/itex] your series converges using that test.
  6. Apr 8, 2007 #5
    yes, so i have

    lim n --> oo ((3x + 4)^(n+1)/ (n +1)) * (n/ (3x +4)^n)

    which simplifies to lim (3x + 4) (n/ (n+1))

    so is it abs value (3x+4) < 1 if it converges? but i dont think i have this right b/c none of the answer choices fit to make this statement true.

    a) 0
    b) 1/3
    c) 2/3
    d) 3/4
    e) 4/3
  7. Apr 9, 2007 #6


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    Science Advisor

    You havent finished yet. You need x, not 3x+4. Saying that |3x+4|< 1 means -1< 3x+4< 1. Now what interval does x lie in? What is the length of that interval? Of course, the "radius" of convergence is half the length of the interval of convergence.
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