Finding Taylor series about some point

In summary, a Taylor series is a way to represent a function as an infinite sum of terms, using its derivatives evaluated at a specific point. This allows for approximating the function's value and understanding its behavior around that point. The process involves finding the derivatives at the given point and plugging them into a general formula, resulting in a simplified finite sum. A Maclaurin series is a special case of a Taylor series where the point of expansion is 0. Taylor series are used in various fields such as physics, engineering, finance, computer graphics, and statistics for approximations, predictions, and modeling.
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JG89
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In this: http://www.math.tamu.edu/~fulling/coalweb/sinsubst.pdf


It says that to find the Taylor series of sin(2x + 1) around the point x = 0, we cannot just substitute 2x+1 into the Maclaurin series for sinx because 2x + 1 doesn't approach a limit of 0 as x approaches 0.

It says we have to find the Taylor series for sinx about x = 1, and then we can substitute 2x + 1 into it.

Why is this?
 
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  • #2
The Taylor series for sinu around 1 will be a series of powers of (u-1). Substitute 2x+1 for u and you will have a series in powers of 2x.
 

What is a Taylor series?

A Taylor series is a mathematical representation of a function as an infinite sum of terms, with each term being a multiple of the function's derivatives evaluated at a specific point.

Why is finding the Taylor series about a point important?

Finding the Taylor series about a specific point allows us to approximate the value of a function at that point, as well as to understand the overall behavior of the function around that point.

What is the process for finding the Taylor series about a point?

The process involves calculating the derivatives of the function at the given point, and then plugging these values into the general formula for a Taylor series. This results in an infinite sum which can be simplified to a finite number of terms for practical use.

What is the difference between a Taylor series and a Maclaurin series?

A Taylor series is a representation of a function around any given point, while a Maclaurin series is a specific type of Taylor series where the point of expansion is 0. In other words, a Maclaurin series is a special case of a Taylor series where the function is centered at 0.

How is a Taylor series used in real-world applications?

Taylor series are used in various fields such as physics, engineering, and finance to approximate complex functions and make predictions. They are also used in computer graphics to create smooth curves and surfaces, and in statistics to model and analyze data.

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