Taylor Expansion

In mathematics, the Taylor expansion (or Taylor series) is a way to represent a function as an infinite sum of terms. It is named after the English mathematician Brook Taylor, who first described it in 1715. The Taylor expansion is used in calculus, analysis, and other areas of mathematics to approximate functions and to solve differential equations.

Formula

The Taylor expansion of a function $f(x)$ around a point $a$ is given by:

$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$

where $f^{(n)}(a)$ is the $n$-th derivative of $f$ evaluated at $a$, and $n!$ denotes the factorial of $n$.

The terms in the Taylor expansion are called the Taylor coefficients. The approximation provided by the Taylor expansion becomes more accurate as more terms are added to the summation.

Example

Let's consider the function $e^x$ and its Taylor expansion around $a=0$. The derivatives of $e^x$ are $e^x$, $e^x$, $e^x$, $\ldots$, and evaluated at $a=0$ they are all equal to $1$. Therefore, the Taylor expansion of $e^x$ around $a=0$ is:

$e^x = \sum_{n=0}^{\infty} \frac{1}{n!}x^n$

This is known as the Maclaurin series of $e^x$.

Applications

The Taylor expansion is a powerful tool in many areas of mathematics and science. For example:

• In physics, the Taylor expansion is used to approximate the behavior of physical systems. For instance, the motion of a pendulum can be approximated using the Taylor expansion of the sine function.
• In engineering, the Taylor expansion is used to approximate the behavior of complex systems. For instance, the response of a control system can be approximated using the Taylor expansion of the transfer function.
• In finance, the Taylor expansion is used to calculate the value of financial derivatives. For instance, the Black-Scholes formula for pricing options is based on the Taylor expansion of the stock price.

Conclusion

The Taylor expansion is a powerful mathematical tool that is used in a variety of fields. It allows us to approximate complex functions and solve differential equations. The Taylor expansion is a fundamental concept in calculus and analysis, and is essential to understanding many areas of modern mathematics and science.

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