Chapter 2: Q14 MP (page 80)

Find the disk of convergence of the series $\sum_{} {(z - 2 i)}^{n} / n$ .

Short Answer

Expert verified

The equation $|z - 2 i| < 1$ represents the equation of a disk having $(0, 2)$ as a center.

Step by step solution

Given Information

The given expression is $\sum_{} \frac{{(z - 2 i)}^{n}}{n}$ .

Definition of Complex Numbers

The numbers that are presented in the form of $x + i y$ where, $' x'$ is real numbers and $' i y'$ is an imaginary number, those numbers are referred to as called Complex numbers.

Step 3:Find the ratio of the series.

Consider $s = \sum_{} \frac{(z - 2 i)}{n}$ be a series

Let $\frac{{(z - 2)}^{n}}{n}$ be $a_{n}$

$a_{n} = \frac{{(z - 2 i)}^{n}}{n}$ …(1)

Replace $n$ by $n + 1$ in equation (1).

$a_{n + 1} = \frac{{(z - 2 i)}^{n + 1}}{(n + 1)}$ …(2)

Find the ratio of $p_{n}$ .

$p_{n} = \frac{(\frac{{(z - 2 i)}^{n + 1}}{(n + 1)})}{(\frac{{(z - 2 i)}^{n}}{n})} = (\frac{{(z - 2 i)}^{n + 1}}{n + 1}) \times (\frac{n}{{(z - 2 i)}^{n}}) p_{n} = (z - 2 i) \times (\frac{n}{n + 1})$

Finding the Limit

Find the limit of $p_{n}$ .

role="math" localid="1658906723093" $p = \lim_{n \to \infty} |(z - 2 i) \times \frac{n}{n + 1}| = \lim_{n \to \infty} |(z - 2 i)| |\frac{n}{n + 1}| =' |(z - 2 i)| \underset{n \to \infty}{l i m} |\frac{1}{1 + \frac{1}{n}}| = |(z - 2 i)|$

The condition $p = \underset{n \to \infty}{l i m} |p_{n}| < 1$ must be valid for this series to converge.

$|z - 2 i| < 1$

Hence the equation $|z - 2 i| < 1$ represents the equation of a disk having as a center.

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Find the disk of convergence of the series $\sum_{} {(z - 2 i)}^{n} / n$ .

Short Answer

Step by step solution

Given Information

Definition of Complex Numbers

Step 3:Find the ratio of the series.

Finding the Limit

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Find the disk of convergence of the series∑(z-2i)n/n.

Short Answer

Step by step solution

Given Information

Definition of Complex Numbers

Step 3:Find the ratio of the series.

Finding the Limit

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Scientific Method Physics

Translational Dynamics

Radiation

Conservation of Energy and Momentum

Physics of Motion

Electrostatics

Study anywhere. Anytime. Across all devices.

Find the disk of convergence of the series $\sum_{} {(z - 2 i)}^{n} / n$ .