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Chapter 10: Problem 79

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

Short Answer

Expert verified
Question: Find a power series with an interval of convergence containing (2,6). Answer: One possible power series with an interval of convergence containing (2,6) is \[ \sum_{n=0}^{\infty} \frac{1}{2^n} (x-4)^n \]

Step by step solution

01

Choose the center of the power series

First, let's choose a center for the power series that is within the interval (2,6). A good choice is the midpoint of the interval, which is at x=4. Therefore, we can create the power series around this point.
02

Define the formula of the power series

Now let's create a power series with its center at x=4, which looks like this: \[ \sum_{n=0}^{\infty} a_n (x-4)^n \] where \(a_n\) are the coefficients we need to determine.
03

Apply the ratio test for convergence

To find the radius of convergence, we will use the ratio test. The ratio test states that if \(\lim_{n\to\infty} |\frac{a_{n+1}}{a_n}| < 1\), then the series converges. Applying the ratio test to our power series, we can get: \[ L =\lim_{n\to\infty} |\frac{a_{n+1}(x-4)^{n+1}}{a_n(x-4)^n}| = |(x-4)|\lim_{n\to\infty} |\frac{a_{n+1}}{a_n}| \]
04

Define the radius of convergence

We want \(L < 1\) for the interval of convergence. This can be rewritten as: \[|x-4| < \frac{1}{\lim_{n\to\infty}|\frac{a_{n+1}}{a_n}|}\] The right side of this inequality represents the radius of convergence, \(R\). Thus, we have the inequality: \[|x-4| < R\]
05

Calculate the endpoints and coefficients

The interval of convergence is between (2, 6), so the distance from the center to each endpoint is 2, meaning \(R = 2\). Now, choose any coefficients \(a_n\) for our power series that satisfy the ratio test, making them constant for simplicity. For example, let \(a_n = \frac{1}{2^n}\). Then we have: \[\lim_{n\to\infty}|\frac{a_{n+1}}{a_n}| = \lim_{n\to\infty}|\frac{\frac{1}{2^{n+1}}}{\frac{1}{2^n}}| = \frac{1}{2} < 1\]
06

Write down the final power series

With the given coefficients, the power series becomes: \[ \sum_{n=0}^{\infty} \frac{1}{2^n} (x-4)^n \] This power series has (2,6) as its interval of convergence, as required in the exercise.

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Most popular questions from this chapter

Teams \(A\) and \(B\) go into sudden death overtime after playing to a tie. The teams alternate possession of the ball and the first team to score wins. Each team has a \(\frac{1}{6}\) chance of scoring when it has the ball, with Team \(\mathrm{A}\) having the ball first. a. The probability that Team A ultimately wins is \(\sum_{k=0}^{\infty} \frac{1}{6}\left(\frac{5}{6}\right)^{2 k}\) Evaluate this series. b. The expected number of rounds (possessions by either team) required for the overtime to end is \(\frac{1}{6} \sum_{k=1}^{\infty} k\left(\frac{5}{6}\right)^{k-1} .\) Evaluate this series.

Determine whether the following statements are true and give an explanation or counterexample. a. To evaluate \(\int_{0}^{2} \frac{d x}{1-x},\) one could expand the integrand in a Taylor series and integrate term by term. b. To approximate \(\pi / 3,\) one could substitute \(x=\sqrt{3}\) into the Taylor series for \(\tan ^{-1} x\) c. \(\sum_{k=0}^{\infty} \frac{(\ln 2)^{k}}{k !}=2\)

Use properties of power series, substitution, and factoring of constants to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. Use the Taylor series. $$(1+x)^{-2}=1-2 x+3 x^{2}-4 x^{3}+\cdots, \text { for }-1 < x < 1$$ $$\left(x^{2}-4 x+5\right)^{-2}$$

The theory of optics gives rise to the two Fresnel integrals $$S(x)=\int_{0}^{x} \sin t^{2} d t \text { and } C(x)=\int_{0}^{x} \cos t^{2} d t$$ a. Compute \(S^{\prime}(x)\) and \(C^{\prime}(x)\) b. Expand \(\sin t^{2}\) and \(\cos t^{2}\) in a Maclaurin series and then integrate to find the first four nonzero terms of the Maclaurin series for \(S\) and \(C\) c. Use the polynomials in part (b) to approximate \(S(0.05)\) and \(C(-0.25)\) d. How many terms of the Maclaurin series are required to approximate \(S(0.05)\) with an error no greater than \(10^{-4} ?\) e. How many terms of the Maclaurin series are required to approximate \(C(-0.25)\) with an error no greater than \(10^{-6} ?\)

The expected (average) number of tosses of a fair coin required to obtain the first head is \(\sum_{k=1}^{\infty} k\left(\frac{1}{2}\right)^{k} .\) Evaluate this series and determine the expected number of tosses. (Hint: Differentiate a geometric series.)
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