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Math Help - Find the Maclaurin series for the function using definition of a Mclaurin series

  1. #1
    Junior Member moonman's Avatar
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    Find the Maclaurin series for the function using definition of a Mclaurin series

    Find the Maclaurin series for the function using definition of a Mclaurin series. Also, find the radius of convergence.

    f(x) = e^{-5x}
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    Junior Member moonman's Avatar
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    Re: Find the Maclaurin series for the function using definition of a Mclaurin series

    This should be the Mclaurin series? \sum^\infty_{n=0} \frac{-5x^n}{n!}

    How do I find the radius of convergence?
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    Re: Find the Maclaurin series for the function using definition of a Mclaurin series

    For power series the simplest way to find the radius of convergence is to use the "ratio test".
    A series of positive numbers, \sum a_n will converge as long as the sequence \frac{a_{n+1}}{a_n} has a limit less than 1. (If that limit is larger than 1, it diverges. If it is equal to 1, we don't know.)
    Here, a_n= \frac{-5x^n}{n!}. Those terms may not be positive but we know that a power series always converges absolutely inside its radius of convergence so we can take the absolute value:
    \left|\frac{-5x^{n+1}}{(n+1)!}\frac{n!}{-5x^n}\right|= \frac{|x|}{n+1}

    What is the limit of that as n goes to infinity? For what values of x is that limit less than 1?
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    Junior Member moonman's Avatar
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    Re: Find the Maclaurin series for the function using definition of a Mclaurin series

    \lim_{n \to \infty} \frac{|x|}{n+1}

    \lim_{n \to \infty} \frac{\frac{|x|}{n}}{\frac{n}{n}+\frac{1}{n}}


    = 0 Not sure I'm doing this right?
    Last edited by moonman; December 9th 2012 at 10:35 AM.
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    Re: Find the Maclaurin series for the function using definition of a Mclaurin series

    e^{-5x} = 1 + (-5x) + \frac{(-5x)^2}{2!} +  \frac{(-5x)^3}{3!} +  \frac{(-5x)^4}{4!} + ... = \sum_{n=0}^\infty \frac{(-5x)^n}{n!}
    Last edited by skeeter; December 9th 2012 at 11:16 AM.
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    Junior Member moonman's Avatar
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    Re: Find the Maclaurin series for the function using definition of a Mclaurin series

    Since the ratio test = 0 then we don't know?
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    Re: Find the Maclaurin series for the function using definition of a Mclaurin series

    Quote Originally Posted by moonman View Post
    Since the ratio test = 0 then we don't know?
    have you used the ratio test to find a radius of convergence before?

    \lim_{n \to \infty} \left| \frac{(-5x)^{n+1}}{(n+1)!} \cdot \frac{n!}{(-5x)^n} \right| < 1

    \lim_{n \to \infty} \left| \frac{-5x}{n+1} \right| < 1

    5|x| \cdot \lim_{n \to \infty} \frac{1}{n+1} < 1

    5|x| \cdot 0 < 1 for all x ... so, what's the radius of convergence?
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    Junior Member moonman's Avatar
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    Re: Find the Maclaurin series for the function using definition of a Mclaurin series

    infinity! Thank you sir!
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