# Quick Question on Taylor Expansions

• StephenD420
In summary: These formulas are useful in approximating the solutions of ordinary and partial differential equations, and they are also useful for numerical integration.In summary, Taylor series expansions are commonly used in various mathematical applications, such as approximating integrals and obtaining accurate finite difference formulas for numerical analysis. They are particularly helpful when evaluating functions that are not easily tabulated or when solving differential equations.
StephenD420
Hello all,

I am a senior physics undergraduate student. I have wondered about the Taylor Expansion for a few years now and just have never bothered to ask. But I will now:

I know the Taylor Expansion goes like:

f(a) + $\frac{f'(a)}{1!}$*(x-a) + $\frac{f''(a)}{2!}$*(x-a)$^{2}$ + $\frac{f'''(a)}{3!}$*(x-a)$^{3}$ + ...

which is the same as $\sum$ $\frac{f^{n}(a)}{n!}$*(x-a)$^{n}$

but how do you know when you use this to approximate a formula? Any problem that my professors have given they have explicitly said to use a Taylor Expansion, but I know there has to be a rule of thumb when to use the Taylor Expansion to approximate a formula.

Any ideas?
Thanks much.
Stephen

You are asking about typical practical applications of Taylor series expansions. Here are two examples:

1. Suppose you need to evaluate the definite integral of a function over a fairly narrow range of integration limits, and the integral of the function is not conveniently tabulated. You can still get an accurate estimate of the integral by expanding in a Taylor series with respect to some point between the two limits of integration, and integrating several terms in the resulting Taylor series.

2. Taylor series expansions are used extensively in numerical analysis to provide accurate finite difference formulas for the derivatives of various orders of a function.

## What is a Taylor Expansion?

A Taylor Expansion is a mathematical series that represents a function as an infinite sum of terms. It is used to approximate a function at a specific point by using the function's derivatives at that point.

## Why are Taylor Expansions useful?

Taylor Expansions are useful because they allow for the approximation of complex functions at a specific point, making it easier to evaluate the function and its derivatives. They also provide a way to estimate the behavior of a function near a given point.

## What is the formula for a Taylor Expansion?

The formula for a Taylor Expansion is f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ..., where f(a) represents the value of the function at a and f'(a), f''(a), etc. represent the derivatives of the function at a.

## What is the difference between a Taylor Series and a Taylor Polynomial?

A Taylor Series is an infinite sum of terms, while a Taylor Polynomial is a finite sum of terms. A Taylor Polynomial is a truncated version of a Taylor Series, and it is often used to approximate a function with a desired degree of accuracy.

## How do I use a Taylor Expansion to approximate a function?

To use a Taylor Expansion, you need to know the function's derivatives at a specific point. Plug in those values into the formula for a Taylor Expansion and simplify the expression to obtain an approximation for the function at that point.

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