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Chapter 9: Problem 63

In general, how does the accuracy of a Taylor polynomial change as the degree of the polynomial increases? Explain your reasoning.

Short Answer

Expert verified
The accuracy of a Taylor polynomial generally increases as the degree of the polynomial increases. This is because higher degree polynomials are able to fit the function more accurately, particularly near the point of expansion. However, increasing the degree indefinitely may lead to overfitting, inaccurately representing the function away from the expansion point.

Step by step solution

01

Understand the Taylor Polynomial

Taylor polynomial is a polynomial of degree n that is used to approximate a function. It is used when the function being studied is too complex to be solved directly. It is based on information available about the function鈥檚 derivatives at a single point.
02

Connection between Degree and Accuracy

The accuracy of a Taylor polynomial typically increases as the degree of the polynomial increases. This is because the higher degree polynomials are capable of fitting the function more accurately, especially in the vicinity of the point of expansion.
03

Understanding the Limit

However, increasing the degree of the Taylor Polynomial will not always increase accuracy. This is due to phenomena like overfitting, where the polynomial starts to fit the noise in the data, or the function behavior in other regions poorly represented by the higher degree.
04

The Trade-off

Therefore there is a trade-off, higher degrees can lead to highly accurate estimates near the point of expansion, but may perform poorly away from that point. Choosing the degree of the Taylor Polynomial thus requires a balance between these factors.

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Most popular questions from this chapter

Using a Binomial Series In Exercises \(17-26,\) use the binomial series to find the Maclaurin series for the function. $$ f(x)=\frac{1}{\sqrt{4+x^{2}}} $$

What can you conclude about the convergence or divergence of \(\sum a_{n}\) for each of the following conditions? Explain your reasoning. (a) \(\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=0\) (b) \(\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=1\) (c) \(\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=\frac{3}{2}\) (d) \(\lim _{n \rightarrow \infty} \sqrt[n]{\left|a_{n}\right|}=2\) (e) \(\lim _{n \rightarrow \infty} \sqrt[n]{\left|a_{n}\right|}=1\) (f) \(\lim _{n \rightarrow \infty} \sqrt[n]{\left|a_{n}\right|}=e\)

Finding a Maclaurin Series In Exercises \(41-44\) , find the Maclaurin series for the function. (See Examples 7 and \(8 . )\) $$ h(x)=x \cos x $$

Proof In Exercises \(13-16,\) prove that the Maclaurin series for the function converges to the function for all \(x .\) $$ f(x)=\cos x $$

Finding a Limit In Exercises \(59-62,\) use the series representation of the function \(f\) to find \(\lim _{x \rightarrow 0} f(x)\) (if it exists). $$ f(x)=\frac{\ln (x+1)}{x} $$
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