Expressing a function as a power series

In summary: This is possible because f(x) can be rewritten as f(x) = 1/(1+x-1), which is equivalent to f(x) = 1/(1+(1-x)). Therefore, the power series for f(x) is the summation from n=0 to infinity of (1-x)^n.
  • #1
jomelmaroma
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Hey guys! Suppose you have a function f(x)=1/2-x which you need to express as a power series. I am familiar with the conventional way of solving its series form, which involves taking out 1/2 from f(x) and arriving with a rational function 1/1-(x/2) which is easy to express as a power series.

I just had an idea: Since I can express f(x) as f(x)=1/(1+(1-x)), does that mean I can take r as 1-x such that the power series is summation from n=0 to infinity (1-x)^n?

Thanks!
 
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  • #2
jomelmaroma said:
Hey guys! Suppose you have a function f(x)=1/2-x which you need to express as a power series. I am familiar with the conventional way of solving its series form, which involves taking out 1/2 from f(x) and arriving with a rational function 1/1-(x/2) which is easy to express as a power series.

I just had an idea: Since I can express f(x) as f(x)=1/(1+(1-x)), does that mean I can take r as 1-x such that the power series is summation from n=0 to infinity (1-x)^n?

Thanks!

Series involving powers of x are called MacLaurin series. Series involving powers of (x-a), where a is constant are called Taylor series. What you are proposing is a Taylor series with a = 1.
 

1. What is a power series?

A power series is a representation of a function as an infinite sum of terms, where each term is a power of the independent variable multiplied by a coefficient. It is written in the form of ∑n=0∞ anxn, where an are the coefficients and x is the independent variable.

2. Why is it useful to express a function as a power series?

Expressing a function as a power series allows us to approximate the value of the function at any point by using a finite number of terms. This can be helpful in solving equations, finding derivatives and integrals, and understanding the behavior of a function.

3. What is the process of expressing a function as a power series?

The process involves finding the coefficients an and the value of x at which the series is centered. This is usually done by using known power series and manipulating them to match the given function. The resulting series may have a finite or infinite number of terms.

4. Can any function be expressed as a power series?

No, not all functions can be represented as a power series. Only functions that are analytic, meaning they have continuous derivatives of all orders, can be expressed as a power series. Examples of non-analytic functions include absolute value, step function, and the floor function.

5. How accurate is the approximation using a power series?

The accuracy of the approximation depends on the number of terms used in the series. The more terms used, the more accurate the approximation will be. However, as the number of terms increases, so does the complexity of the calculation. So, the number of terms should be chosen carefully based on the desired level of accuracy and the computational resources available.

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