Convergence of Taylor Series for Various Functions

In summary, a Taylor series is a mathematical representation of a function that uses an infinite sum of terms to approximate its value at a specific point. Its purpose is to approximate functions that are difficult or expensive to evaluate. The formula for a Taylor series involves the function's derivatives at the point of approximation. It differs from a Maclaurin series in that it can approximate the function at any point, while a Maclaurin series only approximates at 0. The convergence of a Taylor series depends on the smoothness of the function and the point of approximation, which can be determined using the ratio or root test.
  • #1
Berdi
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Homework Statement



For what values of x (or [tex]\theta[/tex] or u as appropriate) do you expect the following Taylor Series to converge? DO NOT work out the series.

[tex]\sqrt{x^{2}-x-2}[/tex] about x = 1/3

[tex]sin(1-\theta^{2}) [/tex] about [tex]\theta = 0[/tex]


[tex]tanh (u) [/tex] about u =1


Homework Equations





The Attempt at a Solution



I'm not to sure where to begin. Taylor series have a radius of convergence where |x-a|< R, wher a is the nearest singularity, so I suppose that's a starting point?
 
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  • #2
For what domain is [tex]\sqrt{x^{2}-x-2}[/tex] defined? It can't converge beyond that.
 

What is a Taylor series?

A Taylor series is a mathematical representation of a function that can be expressed as an infinite sum of terms. It is used to approximate the value of a function at a particular point by using a series of derivatives of the function at that point.

What is the purpose of a Taylor series?

The purpose of a Taylor series is to approximate the value of a function at a particular point, especially when the function cannot be easily evaluated or when it is computationally expensive to do so.

What is the formula for a Taylor series?

The formula for a Taylor series is f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2 + f'''(a)(x-a)^3 + ..., where a is the point at which the function is being approximated and f'(x), f''(x), f'''(x), etc. are the derivatives of the function.

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

A Taylor series is a generalization of the Maclaurin series, where the function is approximated at a point other than 0. A Maclaurin series is a special case of a Taylor series where the point of approximation is 0, making the formula simpler.

How do you determine the convergence of a Taylor series?

The convergence of a Taylor series depends on the function being approximated and the point of approximation. Generally, the series will converge if the function is sufficiently smooth and the point of approximation is within the radius of convergence, which can be determined by the ratio test or the root test.

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