# Series Expansion of 1/Polynomial

• Hepth
In summary, there is no simple way to series expand a function of the form $$\frac{1}{\sum_{n=0}^{\infty} a_n x^n}$$ about the point ##x=0##, such that it can be expressed as another sum ##\sum_n c_n x^n##.
Hepth
Gold Member
Is there a simple way to series expand a function of the form
$$\frac{1}{\sum_{n=0}^{\infty} a_n x^n}$$
about the point ##x=0##, such that it can be expressed as another sum ##\sum_n c_n x^n##?

I tried doing it by taylor expansion but I end up with a sum of sums of products of sums :) and its been too long for me to remember a lot of the more advanced simplifications.

Thanks.

Well, I'd try... $$\sum_{n=0}^\infty b_nx^n = \frac{1}{\sum_{n=0}^\infty a_nx^n}\\ \implies \left(\sum_{n=0}^\infty b_nx^n\right)\left(\sum_{n=0}^\infty a_nx^n\right)=1\\ \implies \sum_{n=0}^\infty \left( \sum_{k=0}^n b_ka_{n-k}\right)x^n = 1\\ \implies \sum_{n=0}^\infty c_n x^n = 1$$... you know what the convolution values are: ##c_0=1, c_{n>0}=0##

If you also know all the ##a_n## you can find the corresponding ##b_n##
If you are lucky, there's a pattern.

mheslep
Thanks!

Glad you like it - I figured you'd kinda got as far as line 3 or I'd have forced you to work it out yourself :)

Note: I didn;t use your notation - so where you wanted ##\small{c_n}##'s I gave you ##\small{b_n}##'s

If the series converges, then it gets easier...
i.e. In the special case where ##a_n=a## is a constant with ##n##, and ##|x|<1##, then the RHS of line #1 converges to ##(1-x)/a## making ##b_0=1/a,\; b_1=-1/a,\; b_{n>1}=0##

In general - the series will have a radius of convergence depending on the value of x (and the series). In physics it is common practice to use the first 2-3 terms only, as an approximation.

Enjoy.

I just needed a general approach other than taking a taylor expansion. It IS physics actually, and what I really have are double summations in both the numerator and denominator over two variables that go to arbitrary order.

So

$$\sum_{ij} c_{ij} x^i y^j = \frac{\sum_{nm} a_{nm} x^n y^m}{\sum_{kl} b_{kl} x^k y^l}$$

And I wanted to find the c's. I know the anm and bkl to any order I really want.

If you keep getting double summations like that you may want to consider if you can get where you want more easily through matrix algebra.

i.e. if ##\vec x## and ##\vec y## are column vectors where the nth element is ##x^{n-1}## and ##y^{n-1}## respectively, then we can make a matrix ##\text{A}## of the coefficients so that: $$\vec y^t A x = \sum_{j=1}^N y^{j-1} \sum_{i=1}^N a_{ij}x^{i-1}$$... you want the powers to start at 0 right?

Then the problem looks like -
... given the following:$$\vec y^t \text{C}\vec x = \frac{\vec y^t\text A \vec x}{\vec y^t \text{B} \vec x}$$... express C in terms of A and B.

... but I don't know the problem you are trying to solve.

## What is a series expansion of 1/polynomial?

A series expansion of 1/polynomial is a mathematical representation of a polynomial function in the form of a sum of infinitely many terms. It is used to approximate the value of a polynomial function at a given point.

## How is a series expansion of 1/polynomial calculated?

A series expansion of 1/polynomial is calculated using the Taylor series method, which involves taking derivatives of the polynomial function at a given point and plugging them into a formula. The more terms that are included in the series, the more accurate the approximation will be.

## What is the purpose of a series expansion of 1/polynomial?

The purpose of a series expansion of 1/polynomial is to approximate the value of a polynomial function at a given point, especially when the exact value is difficult or impossible to calculate. It is also used in various areas of mathematics and science, such as in calculus, physics, and engineering.

## What are some limitations of a series expansion of 1/polynomial?

One limitation of a series expansion of 1/polynomial is that it is only an approximation and may not provide an exact value for the polynomial function. Additionally, the accuracy of the approximation depends on the number of terms included in the series, and it may become increasingly complex to calculate with more terms. It may also not converge for certain values of the polynomial function or at certain points.

## How is a series expansion of 1/polynomial used in real-world applications?

A series expansion of 1/polynomial is used in various real-world applications, such as in physics and engineering, to approximate the behavior of systems and make predictions. It is also used in calculus to solve problems involving polynomial functions and in statistics to fit data to a polynomial curve. Additionally, it is used in computer algorithms and software to perform calculations and simulations.

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