Problem 12 - Thomas Calculus

Question

In Exercises \(1-36\) , (a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely, (c) conditionally? $$ \sum_{n=0}^{\infty} \frac{3^{n} x^{n}}{n !} $$

Step-by-Step Solution

Answer

The series converges absolutely for all \(x\); it does not converge conditionally for any \(x\).

Step 1: Identify the Series Components

The given series is \(\sum_{n=0}^{\infty} \frac{3^{n} x^{n}}{n!}\). This is similar to the expansion of the exponential function \(e^{ax}\). In our case, the expression looks like \(e^{3x}\).

Step 2: Find the Radius of Convergence

The series \(\sum_{n=0}^{\infty} \frac{3^{n} x^{n}}{n!}\) looks similar to the form \(\sum_{n=0}^{\infty} \frac{a^{n} x^{n}}{n!}\), which converges for all real numbers, similar to the exponential function \(e^x\). Thus, the radius of convergence \(R\) is \(\infty\).

Step 3: Determine the Interval of Convergence

Since the radius of convergence is infinite, the interval of convergence is \((-\infty, \infty)\).

Step 4: Check for Absolute Convergence

Textbook series of the form \(\sum_{n=0}^{\infty} \frac{3^{n} |x|^{n}}{n!}\) always converges for all \(x\) because the series \(\sum_{n=0}^{\infty} \frac{a^{n} b^{n}}{n!}\) (such as \(e^{ax}\)) converges absolutely for any real number \(b\).

Step 5: Check for Conditional Convergence

A series is conditionally convergent if it converges, but not absolutely. Since this series is absolutely convergent for all \(x\), there are no values for which it is just conditionally convergent.

Key Concepts

Interval of Convergence

Understanding the interval of convergence is essential when dealing with infinite series. When we talk about the interval of convergence, we refer to the range of values for the variable, usually denoted as \(x\), where the series converges. To put it simply, it's where the magic of infinite addition doesn't blow up to infinity nor does it become undefined.

In the case of the series \(\sum_{n=0}^{\infty} \frac{3^{n} x^{n}}{n !}\), the interval of convergence is determined by finding the radius of convergence first, which here is infinite. This means that the series converges for all real values of \(x\). The interval is thus written as \((-\infty, \infty)\), indicating that no matter what real number you choose for \(x\), the series will converge.

One tip to remember: series that resemble the exponential form \(e^{ax}\) typically have infinite radii of convergence, and hence, their intervals extend from negative to positive infinity.

Absolute Convergence

When we discuss absolute convergence, we're checking if a series behaves nicely enough to converge even when you take the absolute values of its terms. In simpler terms, a series \(\sum a_n\) is absolutely convergent if the "absolute value series” \(\sum |a_n|\) itself converges. This provides a strong form of convergence because, if a series converges absolutely, it certainly converges in the regular sense (