Question

Test each of the following series for convergence. $$\sum \frac{(3+2 i)^{n}}{n !}$$

Short answer

The series \(\frac{(3+2i)^n}{n!}\) converges by the Ratio Test.

Step-by-step solution

Recognize the Type of Series

Notice that the series \(\frac{(3 + 2i)^n}{n!}\) resembles the form of a power series or an exponential series. Recall the general form of an exponential series \(\frac{a^n}{n!}\).

Apply the Ratio Test

To determine the convergence of the series, apply the Ratio Test. The Ratio Test evaluates \(|a_{n+1}/a_n|\) and determines convergence based on its limit. Compute \[ \frac{a_{n+1}}{a_n} = \frac{\frac{(3 + 2i)^{n+1}}{(n+1)!}}{\frac{(3 + 2i)^n}{n!}} = \frac{(3 + 2i)\frac{(3 + 2i)^n}{n!}}{(n+1)\frac{(3 + 2i)^n}{n!}} = \frac{(3+2i)}{n+1} \]

Evaluate the Limit Resulting from the Ratio Test

Now evaluate the limit of \(\frac{|3 + 2i|}{n+1}\) as \(n \to \infty\): \[ \text{lim}_{n \to \infty} \frac{|3 + 2i|}{n+1} = \text{lim}_{n \to \infty} \frac{\sqrt{3^2 + (2)^2}}{n+1} = \text{lim}_{n \to \infty} \frac{\sqrt{13}}{n+1} = 0 \]

Conclude the Convergence

Since the limit is 0, which is less than 1, the series \(\frac{(3+2i)^n}{n!}\) converges by the Ratio Test.

Key concepts

Ratio Test

The Ratio Test is a powerful tool to determine the convergence of a series. It is particularly useful for series involving factorials or exponentials. To apply the Ratio Test, we examine the limit \(\frac{|a_{n+1}|}{|a_n|}\). If this limit is less than 1, the series converges. If it is more than 1, the series diverges. If it equals 1, the test is inconclusive. In our example, we have the series \(\frac{(3+2i)^n}{n!}\). By applying the Ratio Test, we find the limit as \(\frac{|3+2i|}{n+1}\), which simplifies to \(\frac{\sqrt{13}}{n+1}\). As \ n \to \infty \, this limit approaches 0. Since 0 is less than 1, the series converges.

Exponential Series

An exponential series is a series of the form \(\frac{a^n}{n!}\). These series are immensely important in mathematics and frequently arise in various fields such as calculus and complex analysis. In our exercise, we have the series \(\frac{(3 + 2i)^n}{n!}\), which resembles the exponential series. One key property of exponential series \ e^z \ is that it converges for all complex numbers \ z \. This stems from the radius of convergence being infinite. By recognizing our series' form and applying the Ratio Test, we confirm that it, too, converges.

Complex Numbers

Complex numbers extend the idea of one-dimensional numbers to two dimensions, combining real and imaginary parts. A complex number \(3 + 2i\) consists of a real part 3 and an imaginary part 2i, where \ i \ is the square root of -1. Operations on complex numbers follow specific rules, like addition, subtraction, multiplication, and division. In the context of our series, when we calculate \[ |3+2i| = \sqrt{3^2 + 2^2} = \sqrt{13} \], we find the magnitude or absolute value of the complex number. This plays a significant role when applying the Ratio Test and helps us determine convergence by ensuring that terms diminish as \ n \ increases.