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Chapter 8: Q. 5 (page 669)

Explain why ${鈭�}_{k = 0}^{鈭�} {(x - k)}^{k}$ is not a power series.

Short Answer

Expert verified

The base of the factors ${(x - k)}^{k}$ varies with the index.

Step by step solution

01

Step 1. Given information.

Consider the given question,

$f (x) = {鈭�}_{k = 0}^{鈭�} {(x - k)}^{k}$

02

Step 2. Explain the series.

If ${鈭�}_{k = 0}^{鈭�} a_{k}$ is a sequence of real numbers and x be a variable and $x_{0}$ be a real number. Then a power series in $x - x_{0}$ is a series of the form localid="1649342585929" ${鈭� a_{k}}_{k = 0}^{鈭�} {(x - x_{0})}^{k}$

Rewriting the series,

$f {鈭� a_{k}}_{k = 0}^{鈭�} {(x - x_{0})}^{k} = a_{0} + a_{1} (x - x_{0}) + a_{2} {(x - x_{0})}^{2} + a_{3} {(x - x_{0})}^{3} + . . .$

The series of $f (x)$ is not a power series, this is because $x_{0}$ in the series $f (x) = {鈭�}_{k = 0}^{鈭�} {(x - k)}^{k}$ , that is k, is not a constant real number.

We can say that the base of the factors ${(x - k)}^{k}$ varies with the index.

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Most popular questions from this chapter

In Exercises 49鈥�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鈥�48.

$54 . \sqrt[3]{x}, 1$

The second-order differential equation

$x^{2} y^{''} + x y^{'} + (x^{2} - p^{2}) = 0$

where p is a non-negative integer, arises in many applications in physics and engineering, including one model for the vibration of a beaten drum. The solution of the differential equation is called the Bessel function of order p, denoted by $J_{p} (x)$ . It may be shown that $J_{p} (x)$ is given by the following power series in x :

$J_{p} (x) = {鈭�}_{k = 0}^{鈭�} \frac{(- 1)^{k}}{k! (k + p)! 2^{2 k + p}} x^{2 k + p}$

What is the interval of convergence for $J_{1} (x)$ ?

What is meant by the interval of convergence for a power series in $x$ ? How is the interval of convergence determined? If a power series in $x$ has a nontrivial interval of convergence, what types of intervals are possible?

Exercise 64-68 concern with the bessel function.

What is the interval for convergence for $J_{0} (x) ?$

What is $x_{0}$ if $(p, q)$ is the interval of convergence for the power series ${鈭�}_{k = 0}^{鈭�} 鈥� a_{k} {(x 鈭� x_{0})}^{k}$ ?

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