Chapter 2: Q11P (page 59)

Find the disk of convergence for each of the following complex power series.
$\sum_{n = 0}^{\infty} \frac{{(n!)}^{3} z^{n}}{(3 n)!}$

Short Answer

Expert verified

Hence, the required disk of convergence is. $| z | < 27$

Step by step solution

Disk of Convergence

For any power series $\sum a_{n} z^{n}$ where z is a complex numbers, then disk of convergence is given by: . $ρ = \underset{n \to \infty}{l i m} | \frac{z \times n}{n + 1} | = | z |$

Step 2:Find the disk of Convergence

The given power series is: $\sum_{n = 0}^{\infty} \frac{{(n!)}^{3} z^{n}}{(3 n)!}$ , where $a_{n} = \frac{{(n!)}^{3} z^{n}}{(3 n)!}$

Now, let us evaluate the ratio as:

$\begin{matrix} ρ = \lim_{n \to \infty} | \frac{a_{n + 1}}{a_{n}} | \\ = \lim_{n \to \infty} | \frac{\frac{{((n + 1)!)}^{3} z^{n + 1}}{(3 (n + 1))!}}{\frac{{(n!)}^{3} z^{n}}{(3 n)!}} | \\ = \lim_{n \to \infty} | z \cdot \frac{{(n + 1)}^{3} {(n!)}^{3}}{{(n!)}^{3}} \cdot \frac{(3 n)!}{3 (n + 1) (3 n + 2) (3 n + 1) (3 n)!} | \\ = \lim_{n \to \infty} | \frac{z}{3} \cdot \frac{{(n + 1)}^{3}}{(n + 1) (3 n + 2) (3 n + 1)} | \end{matrix}$

On further evaluating as:

$\begin{matrix} ρ = \lim_{n \to \infty} | \frac{z}{3} \cdot \frac{{(n + 1)}^{3}}{(n + 1) (3 n + 2) (3 n + 1)} | < 1 \\ = \frac{| z |}{3} \lim_{n \to \infty} | \frac{{(n + 1)}^{2}}{(3 n + 2) (3 n + 1)} | \\ = \frac{| z |}{3} \lim_{n \to \infty} | \frac{{(1 + \frac{1}{n})}^{2}}{(3 + \frac{2}{n}) (3 + \frac{1}{n})} | \\ = \frac{| z |}{3} | \frac{{(1 + 0)}^{2}}{(3 + 0) (3 + 0)} | = \frac{| z |}{27} \end{matrix}$

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

$\begin{matrix} ρ < 1 \\ \frac{| z |}{27} < 1 \\ | z | < 27 \end{matrix}$

Hence, the required disk of convergence is. $| z | < 27$

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Find the disk of convergence for each of the following complex power series.
$\sum_{n = 0}^{\infty} \frac{{(n!)}^{3} z^{n}}{(3 n)!}$

Short Answer

Step by step solution

Disk of Convergence

Step 2:Find the disk of Convergence

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Find the disk of convergence for each of the following complex power series.∑n=0∞(n!)3zn(3n)!

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Most popular questions from this chapter

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Find the disk of convergence for each of the following complex power series.
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