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Chapter 8: Q 50. (page 670)

Find the radius of convergence for the given series: ${鈭�}_{k = 0}^{鈭�} \frac{k!}{{((k + m)!)}^{2}} x^{k}$

Short Answer

Expert verified

The radius of convergence for the series is $1$ .

Step by step solution

01

Step 1. Given information.

The given power series is ${鈭�}_{k = 0}^{鈭�} \frac{k!}{{((k + m)!)}^{2}}$ .

02

Step 2. Find the radius of convergence.

Let us take $b_{k} = \frac{k!}{{((k + m)!)}^{2}} x^{k}$ therefore $b_{k + 1} = \frac{(k + 1)!}{{((k + 1 + m)!)}^{2}} x^{k + 1}$

Thus, $\lim_{k 鈫� 鈭�} |\frac{b_{k + 1}}{b_{k}}| = \lim_{k 鈫� 鈭�} |\frac{\frac{(k + 1)!}{{((k + 1 + m)!)}^{2}} x^{k + 1}}{\frac{k!}{{((k + m)!)}^{2}} x^{k}}| = \lim_{k 鈫� 鈭�} |(\frac{k + 1}{{(k + 1 + m)}^{2}}) x|$

So, by the ratio test of absolute convergence, the series will converge when $|x| < 1$ .

This implies that $x 鈭� (- 1, 1)$ .

Therefore, the radius of the convergence for the series is $1$ .

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Most popular questions from this chapter

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

What is meant by the interval of convergence for a power series in $x - x_{0}$ ? How is the interval of convergence determined? If a power series in $x - x_{0}$ has a nontrivial interval of convergence, what types of intervals are possible.

Let $a_{k} 鈮� 0$ for each value of $k$ , and let ${鈭�}_{k = 0}^{鈭�} 鈥� a_{k} x^{k}$ 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 ${鈭�}_{k = 0}^{鈭�} 鈥� \frac{1}{a_{k}} x^{k}$ ?

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:

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

Find and graph the first four terms in the sequence of partial sums of $J_{o} (x)$ .

What is a difference between the Maclaurin polynomial of order n and the Taylor polynomial of order n for a function f ?

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