Problem 6 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use the Integral Test to determine the convergence or divergence of each of the following series. $$ \sum_{k=100}^{\infty} \frac{3}{(k+2)^{2}} $$

Step-by-Step Solution

Answer

The series converges.

Step 1: Verify Conditions for the Integral Test

The Integral Test requires a function that is positive, continuous, and decreasing on the interval \([c, \, \infty)\). Here, the function \(f(x) = \frac{3}{(x+2)^2}\) is derived from the series. For \(x \geq 100\), \(f(x)\) is positive, continuous, and decreasing since the power of \((x+2)\) in the denominator is positive. Thus, we can apply the Integral Test.

Step 2: Set Up the Integral

Use the function \(f(x) = \frac{3}{(x+2)^2}\). The integration limits match the series limits, so we consider \(\int_{100}^{\infty} \frac{3}{(x+2)^2} \, dx\).

Step 3: Evaluate the Integral

Calculate the improper integral:\[\int_{100}^{\infty} \frac{3}{(x+2)^2} \, dx = \lim_{b \to \infty} \int_{100}^{b} \frac{3}{(x+2)^2} \, dx\]Use substitution to integrate:Let \(u = x+2\), then \(du = dx\).\[\int \frac{3}{u^2} \, du = -\frac{3}{u} + C\]This evaluates to:\[\lim_{b \to \infty} \left[ -\frac{3}{x+2} \right]_{100}^{b} = \lim_{b \to \infty} \left( -\frac{3}{b+2} + \frac{3}{102} \right)\]

Step 4: Determine the Convergence of the Integral

Evaluate the limit:\[\lim_{b \to \infty} \left( -\frac{3}{b+2} + \frac{3}{102} \right) = 0 + \frac{3}{102} = \frac{1}{34}\]Since the integral converges to \(\frac{1}{34}\), the original series converges by the Integral Test.

Key Concepts

Convergence and Divergence

When studying infinite series, understanding convergence and divergence is very important. A series converges if the sums of its terms approach a specific number as more and more terms are added. If these sums do not approach a specific value, the series is said to diverge.
For the series given in the problem, which is \(\sum_{k=100}^{\infty} \frac{3}{(k+2)^{2}}\), we can use the Integral Test to explore its convergence or divergence. This test uses improper integrals to determine the behavior of the series.
The convergence means each partial sum gets closer to some finite number, while divergence indicates that the series will not stabilize to any particular number.

Improper Integrals

Improper integrals are a type of integral where the interval is infinite, or the integrand has an infinite discontinuity. They extend the concept of integration beyond finite bounds. In this exercise, the integral \\(\int_{100}^{\infty} \frac{3}{(x+2)^2} \, dx\)\ evaluates the behavior of the function at infinity.
To compute improper integrals like this, examine the limit as one of the interval boundaries approaches infinity. If this limit results in a finite number, the integral converges. If the limit is infinite, the integral diverges.
This technique is crucial for the Integral Test because it transforms the question of series convergence into a problem of evaluating an improper integral, as demonstrated in the solution.

Positive and Continuous Functions

Positive and continuous functions play a major role in applying the Integral Test. For a function to be positive, all its values over the specified interval need to be greater than zero. This ensures that the entire series is looking at positive quantities only.
Continuity, on the other hand, means there are no gaps or jumps in the function over its domain. For the function \\(f(x) = \frac{3}{(x+2)^2}\)\, it is positive and continuous for \(x \geq 100\) because dividing by the square of any positive number keeps the result positive, and the function doesn't have any breaks or vertical asymptotes within this interval.

• All values are positive.
• The function smoothly progresses without interruption.

These qualities make the function suitable for the Integral Test.

Decreasing Functions

A function is decreasing over a certain interval if it continually reduces as you move from left to right along the x-axis. A function \\(f(x) = \frac{3}{(x+2)^2}\)\ is decreasing for \(x \geq 100\) because each additional increase in \(x\) makes the denominator larger, lowering the overall value of the fraction.
To use the Integral Test, the function should strictly get smaller and not reverse direction over the interval being examined.

• As \(x\) increases, the fraction \(\frac{3}{(x+2)^2}\) decreases.
• There is a clear downward trend.

This ensures the integrity of the test's conclusions since if the function wasn't regularly decreasing, our results from the improper integral wouldn't necessarily indicate the series' behavior accurately.