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QUESTION: Series

  1. May 23, 2006 #1
    Hi

    Looking at the series

    [tex]\sum \limit_{n=1} ^{\infty} \frac{z^{n+1}}{n(n+1)}[/tex]

    This series has the radius of Convergence R = 1.

    Show that the series

    converge for every [tex]z \in \{w \in \mathbb{C} | |w| \leq 1 \}[/tex]

    And Secondly I need to show that

    [tex]g(z) = \sum \limit_{n=1} ^{\infty} \frac{z^{n+1}}{n(n+1)}[/tex]

    Is continius in [tex]z \in \{w \in \mathbb{C} | |w| \leq 1 \}[/tex]

    Solution:

    (1)

    Since R = 1, then

    [tex]\displaystyle \lim_{n \rightarrow \infty} b_n = \displaystyle \lim_{n \rightarrow \infty} \frac{1}{n(n+1)} = 0[/tex]

    [tex]b_n = \displaystyle \lim_{n \rightarrow \infty} b_n = \displaystyle \lim_{n \rightarrow \infty} \frac{1}{(n+1)(n+1)+1} = b_{n +1} [/tex]

    Therefore converge the [tex]z \in \{w \in \mathbb{C} | |w| \leq 1 \}[/tex]

    (2) Doesn't that follow from (1) ?

    Sincerely Yours
    Hummingbird25
     
    Last edited: May 23, 2006
  2. jcsd
  3. May 23, 2006 #2

    benorin

    User Avatar
    Homework Helper

    no, use the ratio test to determine R.
     
  4. May 23, 2006 #3
    Applying the ratio to the original series

    I get

    [tex] \displaystyle \lim_{n \rightarrow \infty} |\frac{a_{n+1}}{a_n}| \\
    = \lim_{n \rightarrow \infty} |\frac{z^{n+2}}{(n+1)(n+2)} * \frac {n(n+1)}{z^{n+1}}| = |z| < 1[/tex]

    Then do I use a specific test show that the original series ?

    converge for every z \in \{w \in \mathbb{C} | |w| \leq 1 \

    and the is Is continius in [tex]z \in \{w \in \mathbb{C} | |w| \leq 1 \}[/tex]

    Sincerely Yours
    Hummingbird25

     
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