Q12P Find the disk of convergence for... [FREE SOLUTION]

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Chapter 2: Q12P (page 59)

Find the disk of convergence for each of the following complex power series.
${鈭�}_{n = 0}^{鈭�} \frac{{(n!)}^{2} z^{n}}{(2 n)!}$

Short Answer

Expert verified

Hence, the required disk of convergence is.

$| z | < 4$

Step by step solution

Define Disk of Convergence

For any power series $鈭� a_{n} z^{n}$ where z is a complex numbers, then disk of convergence is given by: . $蚁 = \underset{n 鈫� 鈭�}{l i m} | \frac{z 脳 n}{n + 1} | = | z |$

Step 2:Find the disk of Convergence

The given power series is: ${鈭�}_{n = 0}^{鈭�} \frac{{(n!)}^{2} z^{n}}{(2 n)!}$ , where, . $a_{n} = \frac{{(n!)}^{2} z^{n}}{(2 n)!}$

Now, let us evaluate the ratio as:

$\begin{matrix} 蚁 = \lim_{n 鈫� 鈭�} | \frac{a_{n + 1}}{a_{n}} | \\ = \lim_{n 鈫� 鈭�} | \frac{\frac{{((n + 1)!)}^{2} z^{n + 1}}{(2 (n + 1))!}}{\frac{{(n!)}^{2} z^{n}}{(2 n)!}} | \\ = \lim_{n 鈫� 鈭�} | z 鈰� \frac{{(n + 1)}^{2} {(n!)}^{2}}{{(n!)}^{2}} 鈰� \frac{(2 n)!}{2 (n + 1) (2 n + 1) (2 n)!} | \\ = \lim_{n 鈫� 鈭�} | \frac{z}{2} 鈰� \frac{{(n + 1)}^{2}}{(n + 1) (2 n + 1)} | \end{matrix}$

On further evaluating as:

$\begin{matrix} 蚁 = \lim_{n 鈫� 鈭�} | \frac{z}{2} 鈰� \frac{{(n + 1)}^{2}}{(n + 1) (2 n + 1)} | \\ = \frac{| z |}{2} \lim_{n 鈫� 鈭�} | \frac{(n + 1)}{(2 n + 1)} | \\ = \frac{| z |}{2} \lim_{n 鈫� 鈭�} | \frac{(1 + \frac{1}{n})}{(2 + \frac{1}{n})} | \\ = \frac{| z |}{2} | \frac{(1 + 0)}{(2 + 0)} | = \frac{| z |}{4} \end{matrix}$

Now, for the series to be convergent, we have . $蚁 < 1$ So,

$\begin{matrix} 蚁 < 1 \\ \frac{| z |}{4} < 1 \\ | z | < 4 \end{matrix}$

Hence, the required disk of convergence is. $| z | < 4$

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91影视

Find the disk of convergence for each of the following complex power series.
${鈭�}_{n = 0}^{鈭�} \frac{{(n!)}^{2} z^{n}}{(2 n)!}$

Short Answer

Step by step solution

Define Disk of Convergence

Step 2:Find the disk of Convergence

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91影视

Find the disk of convergence for each of the following complex power series.鈭�n=0鈭�(n!)2zn(2n)!

Short Answer

Step by step solution

Define Disk of Convergence

Step 2:Find the disk of Convergence

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Quantum Physics

Torque and Rotational Motion

Astrophysics

Further Mechanics and Thermal Physics

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Waves Physics

Study anywhere. Anytime. Across all devices.

Find the disk of convergence for each of the following complex power series.
${鈭�}_{n = 0}^{鈭�} \frac{{(n!)}^{2} z^{n}}{(2 n)!}$