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Chapter 8: Q. 4 (page 669)

What is a power series in $x - x_{0}$ ?

Short Answer

Expert verified

A power series in $x - x_{0}$ is $\sum_{k = 0}^{\infty} a_{k} {(x - x_{0})}^{k} = a_{0} + a_{1} (x - x_{0}) + a_{2} {(x - x_{0})}^{2} + . . .$

Step by step solution

01

Step 1. Given information.

Consider the given question,

$x - x_{0}$ .

02

Step 2. Consider a power series.

If $\sum_{k = 0}^{\infty} a_{k}$ is a sequence of real numbers and x is a variable and $x_{0}$ be a real number. Then a power series in $x - x_{0}$ is a series of the form localid="1649343968159" $\sum_{k = 0}^{\infty} a_{k} {(x - x_{0})}^{k}$ .

Rewriting the series,

$\sum_{k = 0}^{\infty} a_{k} {(x - x_{0})}^{k} = a_{0} + a_{1} (x - x_{0}) + a_{2} {(x - x_{0})}^{2} + . . .$

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Most popular questions from this chapter

Find the interval of convergence for each power series in Exercises 21–48. If the interval of convergence is finite, be sure to analyze the convergence at the endpoints.

$\sum_{k = 0}^{\infty} {(- 1)}^{k} \frac{k^{2}}{3^{k}} {(x - \frac{1}{2})}^{k}$

Let $f$ be a function with an nth-order derivative at a point $x_{o}$ and let $P_{n} (x) = \sum_{k = 0}^{n} \frac{f^{(k)} (x_{0})}{k!} {(x - x_{0})}^{k}$ . Prove that $f^{(k)} (x_{0}) = P_{n}^{(k)} (x_{0})$ for every non-negative integer $k \leq n$ .

In Exercises 49–56 find the Taylor series for the specified function and the given value of $x_{0}$ . Note: These are the same functions and values as in Exercises 41–48.

role="math" localid="1649406173168" $49 . \cos (x), \frac{π}{2}$

Show that the power series $\sum_{k = 1}^{\infty} \frac{(- 1)^{k}}{k^{2}} x^{k}$ converges absolutely when $x = 1$ and when $x = - 1$ . What does this behavior tell you about the interval of convergence for the series?

Let f be a twice-differentiable function at a point $x_{0}$ . Using the words value, slope, and concavity, explain why the second Taylor polynomial $P_{2} (x)$ might be a good approximation for f close to $x_{0}$ .

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