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I was asked to multiply these two power series expansions: ie. $p(x)q(x)$

$$p(x) = 2(x − 4) − 7(x − 4)^2 + 5(x − 4)^3 + . . .$$

$$q(x) = 4 − 3(x − 4) + (x − 4)^3 + . . .$$

and I was wondering if there's a trick to doing this without distributing and multiplying everything out before adding "like terms". Thank you!

The Dragonborn
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sgasf123
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    Generally, a multiplication formula for $f(x) = \sum_{n=1}^\infty a_n x^n$ and $g(x) = \sum_{n=1}^\infty b_n x^n$ is given by

    $$f(x)g(x) = \sum_{n=1}^\infty c_n x^n \text{, where } c_n = \sum_{i=1}^n a_i b_{n-i} = \sum_{i+j=n}a_ib_j$$

    It might be hard to do it in your case without first writing your series in sigma notation first, though, if there even is a clear pattern. You might just have to do it the hard way otherwise, as far as I know.

     – PrincessEev Jun 26 '20 at 20:24
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    Make a multiplication square, with the top being the coefficients of $(x-4)^k$ in $p$ and the left being the coefficients of $(x-4)^k$ in $q$. The bottom-left-to-upper-right diagonals will have the same coefficients in the product. – Neal Jun 26 '20 at 20:25
  • Arrange the coefficients in a 2 D table and than you multiply/add along the diagonals. – Moti Jun 27 '20 at 01:23

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