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Chapter 10: Problem 4

Suggest a Taylor series and a method for approximating \(\pi.\)

Short Answer

Expert verified
Answer: The approximation of 蟺 is given by the sum of the series: \(\pi \approx 4 \cdot \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots\right)\).

Step by step solution

01

Write down the Taylor series expansion formula

To expand a function as a Taylor series around the point \(x = a\), we can use the following formula: \(f(x) = \sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n,\) where \(f^{(n)}(a)\) denotes the n-th derivative of the function evaluated at the point \(x = a\).
02

Find the derivatives of the arctan(x) function

We need to find the derivatives of the arctan(x) function: \(f(x) = \arctan(x)\) \(f'(x) = \frac{1}{1+x^2}\) \(f''(x) = -\frac{2x}{(1+x^2)^2}\) \(f'''(x) = \frac{6x^2-2}{(1+x^2)^3}\) Observe that in general, for the n-th derivative of arctan(x) function, we will have a function of the form: \(f^{(n)}(x) = \frac{P_n(x)}{(1+x^2)^n}\) where \(P_n(x)\) is a polynomial in x.
03

Expand the arctan(x) function at x=0

Now we need to evaluate these derivatives at x=0: \(f(0) = \arctan(0) = 0\) \(f'(0) = \frac{1}{1+0^2} = 1\) \(f''(0) = -\frac{2 \cdot 0}{(1+0^2)^2} = 0\) \(f'''(0) = \frac{6 \cdot 0^2 - 2}{(1+0^2)^3} = -2\) It appears that only odd powers and odd derivatives of x are nonzero when evaluated at x=0.
04

Expand arctan(x) as a Taylor series

Now we can expand the arctan(x) function using the Taylor series formula: \(\arctan(x) = \sum_{n=0}^{\infty}\frac{f^{(2n+1)}(0)}{(2n+1)!}x^{2n+1} = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots\)
05

Use the relation between arctan and 蟺

As \(\pi/4 = \arctan(1)\), we can plug in x=1 into our Taylor series expansion of arctan(x): \(\pi/4 = \arctan(1) = 1 - \frac{1^3}{3} + \frac{1^5}{5} - \frac{1^7}{7} + \cdots\) To approximate 蟺, we can multiply this series by 4: \(\pi \approx 4 \cdot \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots\right)\)
06

Convergence and approximation

The series converges to 蟺, although it converges slowly. To approximate 蟺, we can take a certain number of terms from the series. The more terms we take, the better the approximation will be. For instance, taking the first 500 terms would give a relatively accurate approximation of 蟺. However, there are other methods to approximate 蟺 that converge faster, but this Taylor series provides a simple way to understand the process.

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Most popular questions from this chapter

Compute the coefficients for the Taylor series for the following functions about the given point a and then use the first four terms of the series to approximate the given number. $$f(x)=1 / \sqrt{x} \text { with } a=4 ; \text { approximate } 1 / \sqrt{3}$$

Local extreme points and inflection points Suppose that \(f\) has two continuous derivatives at \(a\) a. Show that if \(f\) has a local maximum at \(a,\) then the Taylor polynomial \(p_{2}\) centered at \(a\) also has a local maximum at \(a\) b. Show that if \(f\) has a local minimum at \(a,\) then the Taylor polynomial \(p_{2}\) centered at \(a\) also has a local minimum at \(a\) c. Is it true that if \(f\) has an inflection point at \(a,\) then the Taylor polynomial \(p_{2}\) centered at \(a\) also has an inflection point at \(a ?\) d. Are the converses to parts (a) and (b) true? If \(p_{2}\) has a local extreme point at \(a,\) does \(f\) have the same type of point at \(a ?\)

An essential function in statistics and the study of the normal distribution is the error function $$\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^{2}} d t$$ a. Compute the derivative of erf \((x)\) b. Expand \(e^{-t^{2}}\) in a Maclaurin series, then integrate to find the first four nonzero terms of the Maclaurin series for erf. c. Use the polynomial in part (b) to approximate erf (0.15) and erf ( -0.09 ). d. Estimate the error in the approximations of part (c).

a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. In most cases you do not need to use the definition of the Taylor series coefficients. b. If possible, determine the radius of convergence of the series. $$f(x)=\frac{1}{x^{4}+2 x^{2}+1}$$

Find a power series that has (2,6) as an interval of convergence.
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