# Calculating Coefficients of Infinite Power Series

• zetafunction
In summary, the conversation discusses the possibility of obtaining the coefficients of the power series for 1/f(x) from the coefficients of f(x). It is mentioned that this can be achieved through long division for the case when a_0 is not equal to 0, but there is no general formula for this process.
zetafunction
given the infinite power series

$$f(x)= \sum_{n=0}^{\infty}a_ {n}x^{n}$$

if we know ALL the a(n) is there a straight formula to get the coefficients of the b(n)

$$\frac{1}{f(x)}= \sum_{n=0}^{\infty}b_ {n}x^{n}$$

for example from the chain rule for 1/x and f(x) could be obtain some combinatorial argument to get the b(n) from the a(n) ??

Not usually. f(x)=x. There is no power series for 1/f(x).

Take the case a_0 not 0. We can obtain the b_n by *long division* of power series.

## 1. What is a power series?

A power series is a representation of a function as an infinite sum of powers of a variable. It is written in the form of ∑n=0∞ an(x-c)n, where an represents the coefficients, x is the variable, and c is the center of the series.

## 2. How do you calculate the coefficients of an infinite power series?

The coefficients of an infinite power series can be calculated using the formula an = f(n)(c)/n!, where f(n)(c) is the nth derivative of the function at the center c. Alternatively, the coefficients can also be found using the Taylor series expansion of the function.

## 3. What is the importance of calculating coefficients of infinite power series?

Calculating coefficients of infinite power series allows for the approximation of a given function using a simpler polynomial form. This can be useful in various mathematical and scientific applications, such as solving differential equations, evaluating complex functions, and analyzing the behavior of a function near a certain point.

## 4. Are there any techniques for simplifying the calculation of coefficients?

Yes, there are techniques such as the geometric series method, the binomial theorem, and the use of recurrence relations, which can make the calculation of coefficients of infinite power series more efficient and manageable.

## 5. Can the coefficients of an infinite power series be negative?

Yes, the coefficients of an infinite power series can be negative, positive, or zero. This depends on the function being represented and the specific values of the coefficients. In some cases, the coefficients may alternate between positive and negative values.

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