Chapter 1: Problem 10

Use the integral test to find whether the following series converge or diverge. Hint and warning: Do not use lower limits on your integrals (see Problem 16 ). $$\sum_{n=1}^{\infty} \frac{e^{n}}{e^{2 n}+9}$$

Short Answer

Expert verified

The series $\sum_{n=1}^{\infty} \frac{e^{n}}{e^{2 n}+9} $ converges.

Step by step solution

Recognize the series

The given series is $\sum_{n=1}^{\infty} \frac{e^{n}}{e^{2 n}+9}$. To determine whether it converges or diverges, use the integral test.

Define the function for the integral test

Define the function $f(x) = \frac{e^x}{e^{2x} + 9}$. Note that $f(n) = \frac{e^n}{e^{2n} + 9}$, matching the terms of the series.

Determine if the function is positive, continuous, and decreasing

Check that $f(x)$ is positive, continuous, and decreasing for $x \geq 1$. Since the numerator $e^x$ and the denominator $e^{2x} + 9$ are both positive and the denominator grows faster than the numerator, $f(x)$ is positive and decreasing for $x \geq 1$. $f(x)$ is also continuous for $x \geq 1$.

Set up the improper integral

Set up the integral to apply the integral test: \[\int_{1}^{\infty} \frac{e^x}{e^{2x} + 9} \, dx \]

Use substitution for easier integration

Let $u = e^x$, so $du = e^x \, dx$. This transforms the integral: \[\int_{1}^{\infty} \frac{du}{u^2 + 9} \]

Solve the integral

The integral \[\int \frac{du}{u^2 + 9} \] is a standard form where the antiderivative is: \[\int \frac{du}{u^2 + 9} = \frac{1}{3} \arctan\left( \frac{u}{3} \right) + C \] Evaluating the bounds: \[\lim_{{t \to \infty}} \left[ \frac{1}{3} \arctan\left( \frac{u}{3} \right) \right]_{e^1}^{e^t} \]

Evaluate the integral

Substituting the bounds, $\frac{1}{3} [ \arctan\left( \frac{e^t}{3} \right) - \arctan\left( \frac{e^1}{3} \right) ]$. As $t \to \infty$, $\frac{e^t}{3} \to \infty$, hence $\arctan\left( \frac{e^t}{3} \right) \to \frac{\pi}{2}$. So, \[\lim_{t \to \infty} \left[ \frac{1}{3} \left( \frac{\pi}{2} - \arctan\left( \frac{e}{3} \right) \right) \right] \] Which is a finite value.

Conclude the test

Since the improper integral converges to a finite value, the original series $\sum_{n=1}^{\infty} \frac{e^{n}}{e^{2 n}+9} $ converges according to the integral test.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Series Convergence

To determine whether a series converges or diverges, we need to investigate the behavior of its terms. A series $ormalfont\text{\Sigma}_{n=1}^{\infty} a_n$ converges if the sum of its terms approaches a finite limit as more terms are added. In our example, we consider the series $ormalfont\text{\Sigma}_{n=1}^{\infty} \frac{e^{n}}{e^{2 n}+9}$. The integral test helps us determine this behavior using an integral rather than just summing up the terms directly. By converting the problem into a continuous function, we can apply integral calculus to reach our conclusion.

Improper Integrals

Improper integrals are a type of integral where one or both of the limits of integration are infinite, or where the integrand becomes infinite within the limits of integration. For the integral test, we often deal with improper integrals because the series sums terms from 1 to infinity. In this exercise, we had to set up the integral $ormalfont\int_{1}^{\infty} \frac{e^x}{e^{2x} + 9} \, dx$. This falls into the category of improper integrals since the upper limit extends to infinity. We use specific techniques to evaluate such integrals, ensuring that they converge to a finite value.

Substitution Method

The substitution method is a technique used to simplify integrals by changing variables. In this case, we let $ormalfont u = e^x$, which simplifies our integral significantly. Under this substitution, $ormalfont du = e^x \, dx$, transforming our original integral to a new form $ormalfont\int_{1}^{\infty} \frac{du}{u^2 + 9}$. This substitution changes a complex expression into a more manageable one, allowing us to use standard antiderivatives to find the solution.

Antiderivatives

An antiderivative of a function is a function whose derivative is the original function. In simpler terms, it’s the inverse process of differentiation. For instance, the integral $ormalfont\int \frac{du}{u^2 + 9}$ uses a standard antiderivative, which is $ormalfont\frac{1}{3}\arctan\left( \frac{u}{3} \right) + C$. Evaluating this antiderivative at the given limits allows us to determine if the improper integral converges. In this solution, we substitute back to get the actual values at the bounds and ensure the convergence of the original series.