Problem 4 - Differential Equations and Linear Algebra

Question

Determine the radius of convergence of the given power series. $$\sum_{n=0}^{\infty} \frac{2^{n} x^{n}}{n}$$

Step-by-Step Solution

Answer

The radius of convergence for the given power series is \(R = \frac{3}{2}\).

Step 1: Identify the nth term

Let's first identify the nth term of the given power series. We will use this term to determine the limit in the Ratio Test. The nth term of the power series can be written as: $$a_n = \frac{2^n x^n}{n}$$

Step 2: Determine the Ratio Test Limit

Now that we have identified the nth term of the power series, we can determine the limit in the Ratio Test. The limit is given by: $$R = \lim_{n\to\infty} \frac{a_{n}}{a_{n+1}}$$ Substituting the general term \(a_n\) and the term \(a_{n+1} = \frac{2^{n+1} x^{n+1}}{n+1}\): $$R = \lim_{n\to\infty} \frac{\frac{2^n x^n}{n}}{\frac{2^{n+1} x^{n+1}}{n+1}}$$

Step 3: Simplify the Ratio Test Limit

Next, we will simplify the limit by multiplying both numerator and denominator by \(n(n+1) 2^{-n-1} x^{-n-1}\): $$R = \lim_{n\to\infty} \frac{n(n+1) 2^{-n-1} x^{-n-1}\cdot\frac{2^n x^n}{n}}{n(n+1) 2^{-n-1} x^{-n-1}\cdot\frac{2^{n+1} x^{n+1}}{n+1}}$$ Simplifying the limit expression: $$R = \lim_{n\to\infty} \frac{n}{n+1} \cdot \frac{2^{-1}}{x}$$

Step 4: Calculate the Limit

Now, we will determine the limit and find the radius of convergence. Taking the limit as \(n \to \infty\): $$R = \frac{1}{x}\cdot\frac{1}{2}$$ For the power series to converge, the absolute value of this ratio must be less than 1: $$\left| \frac{1}{x}\cdot\frac{1}{2} \right| < 1$$

Step 5: Determine the Radius of Convergence

Finally, we will solve for x to find the interval for which the series converges: $$\frac{1}{2} < |x| < 2$$ This gives us the radius of convergence as: $$R = \frac{3}{2}$$ Therefore, the radius of convergence for the given power series is \(R = \frac{3}{2}\).

Key Concepts

Power Series

A power series is an infinite series of the form \(\sum_{n=0}^\infty a_n x^n\), where \(a_n\) represents the coefficient of the nth term, and x is the variable. Each term in the series is a power of x multiplied by a coefficient.

Understanding power series is essential because they represent functions as infinite polynomials and are widely used in calculus for functions that cannot be represented by finite polynomials. Additionally, they hold the key to understanding various properties of functions, such as derivatives and integrals, through their infinite expansions.

Ratio Test

The Ratio Test is a method used to determine the convergence or divergence of an infinite series. The test states that for a series \(\sum a_n\), one can consider the limit \(\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|\). If this limit is less than 1, the series converges absolutely; if it's greater than 1, the series diverges; and if it equals 1, the test is inconclusive.

In practical terms, the Ratio Test helps us decide whether an infinite process will lead to a finite result, which is a cornerstone concept for working with series in calculus.

Convergence of Series

Convergence of a series refers to the idea that as more and more terms are added together, the sum approaches a finite value. This concept is vital in mathematics because it allows us to use infinite processes to model and solve real-world problems.

There are several tests for convergence, such as the Ratio Test, the Root Test, and the Integral Test, among others. Each test has its own criteria to determine the convergence of series. Understanding when and how to apply these tests is crucial for students as they delve into higher-level calculus and analysis.

Limits in Calculus

Limits are a fundamental concept in calculus. They describe the behavior of a function as it approaches a particular point from both sides. The limit evaluates the tendency of a sequence or function as its variable approaches a specific value or infinity.

Calculating limits is foundational for the study of calculus, as they are used to define derivatives, integrals, and the convergence of sequences and series. The concept of a limit allows us to rigorously define instantaneous rates of change, areas under curves, and the sum of infinitely many terms in a series.