What is the Difference Between Power Series and Taylor Series?
🆚 Go to Comparative Table 🆚The main difference between a power series and a Taylor series lies in their representation and the functions they are associated with. Here are the key differences:
- Definition: A power series is a series of the form $$\sum{n=0}^{\infty} an x^n$$, where $$a_n$$ are constants and $$x$$ is a variable. On the other hand, a Taylor series is a specific type of power series that represents a function using its derivatives at a specific point.
- Coefficients: A power series has constant coefficients, while a Taylor series has coefficients that depend on the derivatives of the function at a point.
- Function Representation: A Taylor series approximates a function locally around a point using its derivatives, while a power series represents a function globally.
- Center: A Taylor series is centered at a specific point, while a power series is more general. Every power series can be written as a Taylor series, but not every power series is a Taylor series.
In summary, a power series is a general expression that can represent a wide range of functions, while a Taylor series is a specific type of power series that approximates a function locally using its derivatives. All Taylor series are power series, but not all power series are Taylor series.
Comparative Table: Power Series vs Taylor Series
Power series and Taylor series are both representations of functions in terms of infinite series. However, there are some differences between the two:
| Power Series | Taylor Series |
|---|---|
| A power series is a series of the form $$\sum{n=1}^{\infty} an x^n$$, where $$a_n$$ is a function of the independent variable $$n$$. | A Taylor series is a specific type of power series that represents a function as an infinite sum of terms involving the function's derivatives at a specific point. |
| Power series can be used for a wide range of functions, while Taylor series are associated with a particular function. | Taylor series are used to approximate functions in numerical analysis and are often employed in calculus to study the behavior of functions. |
| Every Taylor series is a power series, but not every power series is a Taylor series. |
An example of a power series is the geometric series:
$$\sum{n=1}^{\infty} an x^n = \frac{1}{1-x}$$
An example of a Taylor series is the expansion of the sine function:
$$\sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$$
In summary, power series are a more general concept, while Taylor series are a specific type of power series that represents a function in terms of its derivatives at a specific point. Both are used to approximate functions, but Taylor series are more commonly used in calculus and numerical analysis.
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