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Chapter 8: Q. 11 (page 679)

If a function f has a Maclaurin series, what are the possibilities for the interval of convergence for that series?

Short Answer

Expert verified

$The possible value for the interval of convergence for the series are (- ρ, ρ), (- ρ, ρ], [- ρ, ρ) and [- ρ, ρ], here, ρ > 0 .$

Step by step solution

01

Step 1. Given information is:

A function f with Maclaurin series

02

Step 2. Solving for interval

$If f has a Maclaurin series, that is, P_{n} (x) = \sum_{k = 0}^{\infty} \frac{f^{k} (0)}{k!} {(x)}^{k} The possible value for the interval of convergence for the series are (- ρ, ρ), (- ρ, ρ], [- ρ, ρ) and [- ρ, ρ], here, ρ > 0 .$

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Most popular questions from this chapter

Prove that if the power series $\sum_{k = 0}^{\infty} a_{k} x^{k}$ and $\sum_{k = 0}^{\infty} a_{k} x^{2 k}$ have the same radius of convergence $p$ , then $p$ is $0, 1,$ or infinite.

Use an appropriate Maclaurin series to find the values of the series in Exercises 17–22.

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

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:

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

What is the interval of convergence for $J_{p} (x)$ where p is a non-negative integer

Let $a_{k} \neq 0$ for each value of $k$ , and let $\sum_{k = 0}^{\infty} a_{k} x^{k}$ be a power series in $x$ with a positive and finite radius of convergence $p$ . What is the radius of convergence of the power series $\sum_{k = 0}^{\infty} \frac{1}{a_{k}} x^{k}$ ?

In Exercises 23–32 we ask you to give Lagrange’s form for the corresponding remainder, $R_{4} (x)$

$e^{x}$

See all solutions

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