Problem 3 - Calculus Early Transcendentals

Question

Can the Integral Test be used to determine whether a series diverges?

Step-by-Step Solution

Answer

Answer: Yes, the Integral Test can be used to determine whether a series diverges. If the improper integral ∫_{1}^{∞} f(x)dx diverges, then the corresponding series ∑_{n=1}^∞ f(n) also diverges, provided f(x) is positive, continuous, and decreasing for all x ≥ 1. However, if these conditions are not met, the Integral Test cannot be applied.

Step 1: Understanding the Integral Test

The Integral Test is a technique used to determine whether a given series converges or diverges. It is applied to a series ∑_{n=1}^∞ f(n) where f(n) is a positive, continuous, and decreasing function for all n ≥ 1. The test states that if the improper integral ∫_{1}^{∞} f(x) dx converges, then the series ∑_{n=1}^∞ f(n) also converges. Conversely, if the integral diverges, then the series diverges as well.

Step 2: Determining Divergence Using the Integral Test

Yes, the Integral Test can be used to determine whether a series diverges. If we apply the Integral Test to a series and find that the improper integral ∫_{1}^{∞} f(x)dx diverges, then we can conclude that the corresponding series ∑_{n=1}^∞ f(n) also diverges. Keep in mind that for the Integral Test to be applicable, it is necessary that the function f(x) is positive, continuous, and decreasing for all x ≥ 1. If these conditions are not met, then the Integral Test cannot be used to draw conclusions about the convergence or divergence of the series.

Key Concepts

Series Convergence

Understanding series convergence is crucial when dealing with infinite series. A series is simply a sum of terms of a sequence. When we talk about convergence, we're asking whether this sum approaches a specific value as the number of terms goes to infinity. If it does, the series is said to converge; if not, it diverges.

Mathematically, series convergence involves assessing whether the sequence of partial sums converges to a finite number:

• If \ lim_{{N \to \infty}} S_N = L \, where \(S_N\) is the \(N^{th}\) partial sum of the series, the series converges to \(L\).
• If the limit does not exist or is infinite, the series diverges.

The Integral Test offers one method to determine this convergence or divergence by linking it to integral calculus.

Integral Convergence

In the context of the Integral Test, integral convergence involves examining the behavior of an associated improper integral. An improper integral is one where the interval of integration is infinite or the function being integrated becomes unbounded within the interval. In this scenario, the integral usually takes the form \ \int_{{1}}^{{\infty}} f(x) dx \, where \(f(x)\) is continuous, positive, and decreasing for \(x \ge 1\).

• If this integral converges to a specific value, then according to the Integral Test, the corresponding series also converges.
• If the integral diverges (i.e., it approaches infinity or doesn't settle on a specific value), the series will diverge too.

This connection stems from the idea that the integral provides an 'accumulated' sum of function values over an interval, mimicking the behavior of a sum of series terms.

Improper Integrals

Improper integrals play a central role in applying the Integral Test. They differ from standard integrals due to their interval of integration or the function's behavior on this interval. There are two main types:

• Integrals with infinite limits, like \ \int_{{1}}^{{\infty}} f(x) dx \.
• Integrals with unbounded functions within the integration range.

To evaluate improper integrals, we often take limits. For the integral from 1 to infinity, we approach it by calculating \\[ \lim_{{b \to \infty}} \int_{{1}}^{{b}} f(x) dx \]. If this limit is a finite number, the integral converges.

Remember, improper integrals align with the behavior of series. Their convergence or divergence offers direct insight into the nature of the series we are studying through the Integral Test. This method is powerful but requires the function to satisfy specific conditions like being positive, continuous, and decreasing over the interval in question.