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Math Help - generating functions

  1. #1
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    generating functions

    determine the sequence generated by (6-2x)/(1+2x-15x^2)
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by stumped765 View Post
    determine the sequence generated by (6-2x)/(1+2x-15x^2)
    What have you tried? Have you tried decomposing this rational function? Once you have this becomes an elementary application of the geometric series.
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  3. #3
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    yah i got (6-2x)/(5x+1) *(6x-2)/(3x-1)
    but I guess im confused about what the numerator tells me about the sequence like i know what to do if it was jsut 1/(5x+1) or 1/(3x-1) but the numerator throws me for a loop.
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by stumped765 View Post
    yah i got (6-2x)/(5x+1) *(6x-2)/(3x-1)
    but I guess im confused about what the numerator tells me about the sequence like i know what to do if it was jsut 1/(5x+1) or 1/(3x-1) but the numerator throws me for a loop.
    Use partial fractions decomposition. It should give you that

    \frac{6-3x}{-(3x-1)(5x+1)}=\frac{33}{8(5x+1)}-\frac{15}{8(3x-1)}

    Now use that \frac{1}{1-z}=\sum_{n=0}^{\infty}z^n

    IF this is indeed what you are looking for the first term should give that \sigma_n=\frac{33}{8}(-1)^n5^n. Now find \delta_n for the second one. And your function will be the generator of \sigma_n+\delta_n
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