Problem 59 - Thomas Calculus

Question

In Exercises \(35-64\) , use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer. $$ \int_{1}^{\infty} \frac{e^{x}}{x} d x $$

Step-by-Step Solution

Answer

The integral \( \int_{1}^{\infty} \frac{e^x}{x} \, dx \) diverges by the Direct Comparison Test.

Step 1: Identify the Type of Improper Integral

The given integral \( \int_{1}^{\infty} \frac{e^x}{x} \, dx \) is an improper integral of type 1 because the upper limit of integration is infinity. This requires us to determine if the integral converges or diverges.

Step 2: Use Direct Comparison Test Strategy

To apply the Direct Comparison Test, we need to find a function \( g(x) \) such that for sufficiently large \( x \), \( 0 \leq \frac{e^x}{x} \leq g(x) \). A potential candidate is \( g(x) = e^x \).

Step 3: Evaluate Comparison Integral

Evaluate the integral \( \int_{1}^{\infty} e^x \, dx \). This integral is divergent because \( \lim_{b \to \infty} \int_{1}^{b} e^x \, dx = \lim_{b \to \infty} (e^b - e) = \infty \).

Step 4: Conclude Divergence via Direct Comparison Test

Since \( \frac{e^x}{x} \leq e^x \) for \( x \geq 1 \) and the integral of \( e^x \) diverges, by the Direct Comparison Test, the original integral \( \int_{1}^{\infty} \frac{e^x}{x} \, dx \) also diverges.

Key Concepts

Direct Comparison Test

The Direct Comparison Test is a useful tool in mathematics to determine if an improper integral converges or diverges. It involves comparing the function of interest to another function whose behavior is known.

To use this test, follow these steps:

• Identify the function you want to integrate, let's call it \( f(x) \).
• Find a simpler function \( g(x) \) such that for all \( x \) in the interval, \( 0 \leq f(x) \leq g(x) \).
• If the integral of \( g(x) \) is known to converge and \( f(x) \leq g(x) \), then the integral of \( f(x) \) also converges.
• If the integral of \( g(x) \) diverges and \( f(x) \geq g(x) \), then the integral of \( f(x) \) also diverges.

For example, consider the integral \( \int_{1}^{\infty} \frac{e^x}{x} \, dx \).
We choose \( g(x) = e^x \), because we know that its integral diverges. Since \( \frac{e^x}{x} \leq e^x \), by the Direct Comparison Test, our original integral also diverges.

Convergence and Divergence

Convergence and divergence describe whether an improper integral reaches a finite limit (converges) or not (diverges) as the range of integration extends to infinity or approaches a point of discontinuity.

An integral \( \int_{a}^{b} f(x) \, dx \) is convergent if the area under the curve \( f(x) \) between points \( a \) and \( b \) adds up to a finite number. It's divergent if this area is infinite.

This understanding is crucial:

• If an integral converges, the function's effect diminishes as the range expands.
• If an integral diverges, the function's effect continues to grow without bound.

In the example of \( \int_{1}^{\infty} \frac{e^x}{x} \, dx \), since the integral doesn't add up to a finite value as \( x \to \infty \), it diverges. Recognizing convergence and divergence helps determine when a function can be approximated within an acceptable error margin.

Limit Comparison Test

The Limit Comparison Test is similar to the Direct Comparison Test but uses limits to compare the behavior of two functions at infinity or near a point of discontinuity.

To apply this test, consider:

• Select two functions \( f(x) \) and \( g(x) \). The function \( g(x) \) should have a known convergence or divergence behavior.
• Compute the limit \( L = \lim_{x \to \infty} \frac{f(x)}{g(x)} \).

If \( L \) is a positive finite number, both integrals \( \int f(x) \, dx \) and \( \int g(x) \, dx \) will either both converge or both diverge. This approach is particularly handy when the Direct Comparison Test is difficult to apply because it's not clear how a function behaves compared to another. Using a simple example:
If you have \( \int_{1}^{\infty} \frac{e^x}{x} \, dx \), you might choose \( g(x) = e^x \). As seen in the Direct Comparison Test, direct limits aren't necessary due to clear comparative behaviors; however, this test would offer another tool should the functions be less straightforward to compare.