Question

Use the Integral Test to determine the convergence of the given series. $$\sum_{n=1}^{\infty} \frac{1}{2^{n}}$$

Short answer

The series \( \sum \frac{1}{2^n} \) converges by the Integral Test.

Step-by-step solution

Identify the Function for the Integral Test

The Integral Test requires us to identify a continuous, positive, decreasing function \( f(x) \) such that \( f(n) = \frac{1}{2^n} \). Here \( f(x) = \frac{1}{2^x} \) is a suitable function since it is continuous, positive, and decreases for all \( x \geq 1 \).

Set up the Improper Integral

To apply the Integral Test, we set up the improper integral \( \int_{1}^{\infty} \frac{1}{2^x} \, dx \). This integral will help us determine the convergence of the series.

Evaluate the Improper Integral

Find the antiderivative of \( \frac{1}{2^x} \). We use the substitution \( u = 2^x \), which gives \( du = 2^x \ln 2 \, dx \) or \( dx = \frac{du}{u \ln 2} \). Thus, the integral becomes \( \int \frac{1}{2^x} dx = \int \frac{1}{u} \frac{1}{\ln 2} \, du = \frac{\ln u}{\ln 2} + C \).

Evaluate the Definite Integral

Evaluate \( \int_{1}^{\infty} \frac{1}{2^x} \, dx \) using the antiderivative: \[ \int_{1}^{\infty} \frac{1}{2^x} \, dx = \left[-\frac{1}{\ln(2)} \cdot 2^{-x}\right]_{1}^{\infty} \]Evaluating this, we get: \[ = \lim_{b \to \infty} \left( -\frac{1}{\ln(2)} \cdot 2^{-b} + \frac{1}{\ln(2)} \cdot 2^{-1} \right) \]\[ = \lim_{b \to \infty} \left( 0 + \frac{1}{2\ln(2)} \right) = \frac{1}{2\ln(2)} \]This is a finite value.

Conclude with the Integral Test

Since the improper integral \( \int_{1}^{\infty} \frac{1}{2^x} \, dx \) converges to a finite number, the series \( \sum_{n=1}^{\infty} \frac{1}{2^{n}} \) also converges by the Integral Test.

Key concepts

Convergence of Series

Understanding whether a series converges or diverges is crucial in mathematical analysis. A series \( \sum_{n=1}^{\infty} a_n \) converges if the sum approaches a specific finite value as more terms are added. In contrast, it diverges if the sum either grows indefinitely or oscillates without settling at a particular value. The Integral Test is a powerful tool to assess the convergence of series. It compares the series to an integral of a positive, continuous, and decreasing function. This comparison provides insight into whether the series reaches a finite number. If the related improper integral converges to a finite value, so does the series.

Improper Integral

An improper integral involves integrating a function over an infinite interval or an interval where the function becomes unbounded. In the context of the Integral Test, we set up an improper integral from 1 to infinity. For the function \(f(x) = \frac{1}{2^x}\), this improper integral is \(\int_{1}^{\infty} \frac{1}{2^x} \, dx\). The evaluation of this integral helps us determine the behavior of the series \(\sum_{n=1}^{\infty} \frac{1}{2^n}\). Calculating if this integral results in a finite number is critical since it directly informs us about the convergence or divergence of the series.

Substitution Method

The substitution method is a technique used to simplify the integration process, especially for functions that involve complex expressions or exponents. In this example, we use the substitution \( u = 2^x \) to find the antiderivative of \( \frac{1}{2^x} \). After substitution, the integral becomes easier to handle and transforms into an expression that can be integrated using standard methods. This method is particularly helpful when dealing with exponential functions as it simplifies the corresponding integrals significantly.

Exponential Functions

Exponential functions, characterized by expressions like \( a^x \), play a key role in many mathematical contexts, particularly in calculus and series. These functions have unique properties, such as continuous growth or decay, which makes them particularly interesting for analysis. In this exercise, the function \( \frac{1}{2^x} \) exhibits exponential decay, meaning it decreases rapidly as \( x \) increases. This property ensures that the series \( \sum_{n=1}^{\infty} \frac{1}{2^n} \) and its corresponding improper integral \( \int_{1}^{\infty} \frac{1}{2^x} \, dx \) are well-suited to analysis via the Integral Test, providing a pathway to determining convergence.