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Math Help - Complex power series help

  1. #1
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    Complex power series help

    Hello everyone,
    Firstly, I am new and my apologies if my questions are buried somewhere in the thread heap already.

    I am taking a complex analysis course, and well, its 2complex4me! So here are my questions...

    I need to find the radius of convergence of the following complex power series: (so naturally z is in C here).

    My apologies for not using nice looking math typeset, but I dont know the code for it.

    The mth order Bessel function:
    Jm(z) = sum[ {((-1)^n) ((z/2)^(m+2n))} / {n!(n+m)!} , n=0, n=infinity ]

    Also

    sum[ z^n!, n=0, n=inf ]

    And

    sum[ (n+a^n)z^n, n=0, n=inf ]

    I will undoubtedly have more questions as I progress in the course, but if someone can help me out here, I would be very greatful!

    Thank you, Damian.
    Last edited by 2complex4me; March 30th 2008 at 06:06 AM. Reason: omission
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  2. #2
    Behold, the power of SARDINES!
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    Quote Originally Posted by 2complex4me View Post
    Hello everyone,
    Firstly, I am new and my apologies if my questions are buried somewhere in the thread heap already.

    I am taking a complex analysis course, and well, its 2complex4me! So here are my questions...

    I need to find the radius of convergence of the following complex power series: (so naturally z is in C here).

    My apologies for not using nice looking math typeset, but I dont know the code for it.

    The mth order Bessel function:
    Jm(z) = sum[ {((-1)^n) ((z/2)^(m+2n))} / {n!(n+m)!} , n=0, n=infinity ]

    Also

    sum[ z^n!, n=0, n=inf ]

    And

    sum[ (n+a^n)z^n, n=0, n=inf ]

    I will undoubtedly have more questions as I progress in the course, but if someone can help me out here, I would be very greatful!

    Thank you, Damian.

    Try the ratio test

    \lim_{n_ \to \infty}\left | \frac{a_{n+1}}{a_n}\right |=L

    if L < 1 the series converges absolutely
    if L > 1 the series divierges
    if L =1 or the limit doesnt exist the test gives no info


    He is an example

    \lim_{n \to \infty} \left |\frac{\overbrace{\frac{(-1)^{n+1}(\frac{z}{2})^{m+2(n+1)}}{(n+1)!(n+1+m)!}}  ^{a_{n+1}}}{ \underbrace{\frac{(-1)^n (\frac{z}{2})^{m+2n}}{(n)!(n+m)!}}_{a_n}} \right |

    Alot of algebra

    =\lim_{n \to \infty} \left| \frac{z^2}{(1+n)(n+1+m)} \right| =0

    So it converges for all values of z the radius is infinity.

    Yeah

    Good luck
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  3. #3
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    Thanks Empty,
    I'll have a crack at that.
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