Chapter 2: Problem 8

Find the disk of convergence for each of the following complex power series. $$\sum_{n=1}^{\infty} \frac{z^{2 n}}{(2 n+1) !}$$

Short Answer

Expert verified

The series converges for all finite z. The disk of convergence is the entire complex plane.

Step by step solution

Identify the given series

The given complex power series is the series the series the series the series the series the series \[\sum_{n=1}^{\infty} \frac{z^{2n}}{(2n+1)!}\]

Use the ratio test for convergence

Using the ratio test, compute the limit:\[\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \] for the given series. Note that a_n = \frac{z^{2n}}{(2n+1)!}.

Compute the ratio of consecutive terms

Calculate the ratio \[ \frac{a_{n+1}}{a_n} = \frac{\frac{z^{2(n+1)}}{(2(n+1)+1)!}}{\frac{z^{2n}}{(2n+1)!}} \] and simplify.

Simplify the ratio

Simplify the fraction to get:\[ \left| \frac{z^{2(n+1)}}{(2(n+1)+1)!} \times \frac{(2n+1)!}{z^{2n}} \right| = \left| z^2 \times \frac{1}{(2n+3)(2n+2)} \right| \]

Take the limit

Take the limit as n approaches infinity:\[ \lim_{n \to \infty} \left| z^2 \times \frac{1}{(2n+3)(2n+2)} \right| = 0 \] Thus, the limit is 0 for any value of z.

Conclude the disk of convergence

Since the radius of convergence R is such that the limit is less than 1 for all |z| < R, and since the limit is always 0, the series converges for all finite z.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ratio Test

The ratio test helps determine the convergence of a series. To use it, we examine the limit of the ratio of successive terms in the series. Specifically, for a series $\sum_{n=0}^{\infty} a_n$, we compute: \[ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|. \] If the limit is less than 1, the series converges. If it's more than 1, the series diverges. If it equals 1, the test is inconclusive.

Disk of Convergence

In complex analysis, the disk of convergence tells us where a series converges. For a power series $\sum_{n=0}^{\infty} a_n(z - z_0)^n$, it converges in a region called a disk centered at $z_0$ with radius $R$, which is the distance within which the series sums to a finite value.
Using the ratio test, if $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\ = L < 1$ for some radius $R$, the series converges within $|z - z_0| < R$. For the given series $\sum_{n=1}^{\infty} \frac{z^{2n}}{(2n+1)!}$, the disk of convergence is determined by evaluating this limit.

Power Series Expansion

A power series is an infinite series of the form $\sum_{n=0}^{\infty} a_n z^n$. It's a way to represent functions as sums of simpler terms. Power series help in analyzing functions near specific points.
For example, the series $\sum_{n=1}^{\infty} \frac{z^{2n}}{(2n+1)!}$ is an expansion involving powers of $z$. Each term $\frac{z^{2n}}{(2n+1)!}$ includes coefficients and powers of $z$. Power series are key in studying complex functions analytically.

Complex Analysis

Complex analysis governs the study of functions of complex variables. It involves analysis, integration, and series expansions. One fundamental aspect is understanding the convergence of power series in the complex plane.
For our series $\sum_{n=1}^{\infty} \frac{z^{2n}}{(2n+1)!}$, complex analysis tools help determine where it sums to finite values. Using methods like the ratio test, we identify convergence regions (disks) in the complex plane, essential for applying functions correctly in areas like physics and engineering.