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Chapter 8: Q. 61 (page 671)

Let $\sum_{k = 0}^{\infty} a_{k} x^{k}$ be a power series in $x$ with a finite radius of convergence $p$ . Prove that if the series converges absolutely at either $\pm p$ , then the series converges absolutely at the other value as well.

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Step 1. Given information.

given,

$\sum_{k = 0}^{\infty} a_{k} x^{k}$

Step 2.

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Most popular questions from this chapter

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

What is the definition of an odd function? An even function?

What is Lagrange’s form for the remainder? Why is Lagrange’s form usually more useful for analyzing the remainder than the definition of the remainder or the integral provided by Taylor theorem?

The second-order differential equation

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

where p is a nonnegative 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:

$J_{p} (x) = \sum_{k = 0}^{\infty} \frac{(- 1)^{k}}{k! (k + p)! 2^{2 k + p}} x^{2 k + p}$

Graph the first four terms in the sequence of partial sums of $J_{1} (x)$ .

Find the interval of convergence for power series: $\sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{(2 k + 1)!} x^{2 k + 1}$

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Let $\sum_{k = 0}^{\infty} a_{k} x^{k}$ be a power series in $x$ with a finite radius of convergence $p$ . Prove that if the series converges absolutely at either $\pm p$ , then the series converges absolutely at the other value as well.

Short Answer

Step by step solution

Step 1. Given information.

Step 2.

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Let ∑k=0∞ akxkbe a power series in xwith a finite radius of convergence p. Prove that if the series converges absolutely at either ±p, then the series converges absolutely at the other value as well.

Short Answer

Step by step solution

Step 1. Given information.

Step 2.

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Pure Maths

Geometry

Calculus

Decision Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Let $\sum_{k = 0}^{\infty} a_{k} x^{k}$ be a power series in $x$ with a finite radius of convergence $p$ . Prove that if the series converges absolutely at either $\pm p$ , then the series converges absolutely at the other value as well.