       Re: symbolic division of series

• To: mathgroup at smc.vnet.net
• Subject: [mg113169] Re: symbolic division of series
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Sat, 16 Oct 2010 13:11:51 -0400 (EDT)
• References: <201010151752.NAA23522@smc.vnet.net>

```Here is an example that answer your question. Suppose you represent f =
and g as follows:

f = Sum[a[i] x^i, {i, 1, 10}] + O[x]^11;
g = Sum[b[i] x^i, {i, 1, 10}] + O[x]^11;

In other words, you know the first 10 coefficients in the Taylor =
expansion of f and g at 0. Then you can get:

SeriesCoefficient[f/g,5]
(6)/b(1)-(a(5) b(2))/b(1)^2+(a(4) (b(2)^2/b(1)^2-b(3)/b(1)))/b(1)+(a(3) =
(-(b(2)^3/b(1)^3)+(2 b(3) b(2))/b(1)^2-b(4)/b(1)))/b(1)+(a(2) =
(b(2)^4/b(1)^4-(3 b(3) b(2)^2)/b(1)^3+(2 b(4) =
b(2))/b(1)^2+b(3)^2/b(1)^2-b(5)/b(1)))/b(1)+(a(1) (-(b(2)^5/b(1)^5)+(4 =
b(3) b(2)^3)/b(1)^4-(3 b(4) b(2)^2)/b(1)^3-(3 b(3)^2 b(2))/b(1)^3+(2 =
b(5) b(2))/b(1)^2+(2 b(3) b(4))/b(1)^2-b(6)/b(1)))/b(1)

Of course this will only work for the coefficients that can be =
determined form the given information. So SeriesCoefficient[f/g, 9] =
works fine but

SeriesCoefficient[f/g, 10]

Indeterminate

Andrzej Kozlowski

On 15 Oct 2010, at 19:52, Leslaw Bieniasz wrote:

>
>
> Hi,
>
> Suppose that I have two series expansions (Taylor or asymptotic
> expansions) for functions f(x) and g(x). This means I know the =
formulae
> for the series coefficients. Is there any way to use MATHEMATICA
> to obtain symbolically the formulae for the coefficients
> of the analogous series expansion of the ratio f(x)/g(x) ?
> I need a possibly large number of the coefficients of such an =
expansion,
> expressed as functions of the coefficients for f(x) and g(x).
>
> Leslaw
>
>

```

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