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Math Help - Properties of Power Series

  1. #1
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    Smile Properties of Power Series

    Hi!

    I'm taking DFQ and we are learning about solving DE's using the series method.
    I was wondering what the some properties for multiplying 2 power series' are?

    Specifically I need to know how to square an infinite power series. I know I cannot just take the square into the inside of the series, but I cannot find anything online about a formula or method.

    Thanks!
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  2. #2
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    Re: Properties of Power Series

    Quote Originally Posted by beatrixb View Post
    Hi!

    I'm taking DFQ and we are learning about solving DE's using the series method.
    I was wondering what the some properties for multiplying 2 power series' are?

    Specifically I need to know how to square an infinite power series. I know I cannot just take the square into the inside of the series, but I cannot find anything online about a formula or method.

    Thanks!
    Follow the same rules of expanding that you already know, i.e. first term x everything in the second bracket + second term x everything in the second bracket + third term x everything in the second bracket + ...
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    Re: Properties of Power Series

    Quote Originally Posted by Prove It View Post
    Follow the same rules of expanding that you already know, i.e. first term x everything in the second bracket + second term x everything in the second bracket + third term x everything in the second bracket + ...
    Ahh okay so if it were two different series' (A & B) I would get A^2+2AB+B^2?
    And with only one series (A) it is A^2+2A^2+A^2 which would then equal 4A^2?
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    Re: Properties of Power Series

    Quote Originally Posted by beatrixb View Post
    Ahh okay so if it were two different series' (A & B) I would get A^2+2AB+B^2?
    And with only one series (A) it is A^2+2A^2+A^2 which would then equal 4A^2?
    No, I mean

    \displaystyle (a_1 + a_2 + a_3 + \dots)(b_1 + b_2 + b_3 + \dots)\\ = a_1(b_1 + b_2 + b_3 + \dots) + a_2(b_1 + b_2 + b_3 + \dots) + a_3(b_1 + b_2 + b_3 + \dots) + \dots
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    Re: Properties of Power Series

    Quote Originally Posted by beatrixb View Post
    Hi!

    I'm taking DFQ and we are learning about solving DE's using the series method.
    I was wondering what the some properties for multiplying 2 power series' are?

    Specifically I need to know how to square an infinite power series. I know I cannot just take the square into the inside of the series, but I cannot find anything online about a formula or method.

    Thanks!
    The formula for the product of two 'infinite sums' is due to Cauchy...

    \sum_{n=0}^{\infty} a_{n}\ \sum_{k=0}^{\infty} b_{k} = \sum_{n=0}^{\infty} \sum_{k=0}^{n} a_{k}\ b_{n-k} (1)

    Kind regards

    \chi \sigma
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