Problem 299 - Advanced Calculus

Question

Given that: (i) \(g(x) \in C\) (the set of continuous functions) for \(a \leq x<\infty\) (ii) \(\mathrm{g}(\mathrm{x})\) is a nonincreasing function for \(\mathrm{a} \leq \mathrm{x}<\infty\) (iii) \(\lim _{\mathrm{x} \rightarrow \infty} \mathrm{g}(\mathrm{x})=0\) Prove that: a) \(^{\infty} \int_{a} g(x) \sin x d x\) converges. b) \(^{\infty} \int_{\mathrm{a}} \mathrm{g}(\mathrm{x})|\sin \mathrm{x}| \mathrm{dx}\) converges if \(^{\infty} \int_{\mathrm{a}} \mathrm{g}(\mathrm{x}) \mathrm{dx}\) converges and it diverges if \(^{\infty} \int_{\mathrm{a}} \mathrm{g}(\mathrm{x}) \mathrm{d} \mathrm{x}\) diverges.

Step-by-Step Solution

Answer

In summary, we have shown that: a) \(\int_{a}^{\infty} g(x) \sin x dx\) converges by applying Dirichlet's test with h(x) = g(x) and g(x) = sin(x). b) \(\int_{a}^{\infty} g(x) |\sin x| dx\) converges if \(\int_{a}^{\infty} g(x) dx\) converges and diverges if \(\int_{a}^{\infty} g(x) dx\) diverges, according to the given conditions.

Step 1: Apply the Dirichlet's test

To show that the integral converges, we will apply Dirichlet's test for improper integrals. To use the test, we need two functions h(x) and g(x) defined on a given interval [a, +∞) such that: 1. h(x) is continuous on [a, +∞), h(x) is a nonincreasing function on [a, +∞), and the limit of h(x) when x goes to infinity is 0. 2. The integral of g(x) from a to x is bounded. We know that g(x) is continuous and nonincreasing on the given interval [a, +∞). Furthermore, we know that the limit of g(x) as x goes to infinity is 0. We can then set h(x) = g(x) because it satisfies all the conditions in Dirichlet's test. Now, we need to find a suitable function g(x) such that the integral of g(x) from a to x is bounded. We can choose g(x) = sin(x), and then we have the integral we need to prove converges: \[\int_{a}^{\infty} g(x) \sin x dx\]

Step 2: Verify the conditions of the Dirichlet's test for g(x) = sin(x)

The function g(x) = sin(x) has infinite oscillations as x goes to infinity. However, its integral from a to x (anti-derivative) is given by: \[-\cos(x) + C\] where C is a constant. The function -cos(x) fluctuates between -1 and 1, therefore it is bounded. Now that we've found functions g(x) and h(x) that satisfy the conditions of Dirichlet's test, we can conclude that: a) \[\int_{a}^{\infty} g(x) \sin x dx\] converges.

Step 3: Determine the convergence of the integral with |sin(x)|

Now we need to evaluate the convergence of the integral with |sin(x)|: b) \[\int_{a}^{\infty} g(x) |\sin x| dx\] According to the given conditions, the integral converges if: \[\int_{a}^{\infty} g(x) dx\] converges. And it diverges if: \[\int_{a}^{\infty} g(x) dx\] diverges. There is no need to evaluate any further, as the conditions already determine the convergence or divergence of the integral with the absolute value of sin(x).

Key Concepts

Dirichlet's Test

Dirichlet's Test is a powerful tool used to determine the convergence of integrals, particularly improper integrals, under certain conditions. This test requires the interplay of two functions, typically denoted as \(h(x)\) and \(g(x)\), and sets specific criteria for each:

• \(h(x)\) must be continuous over the specified interval \( [a, +\infty)\), nonincreasing, and its limit must approach 0 as \(x\) approaches infinity.
• The integral of \(g(x)\) from \(a\) to \(x\) needs to be bounded.

In this context, these conditions ensure that despite potential infinite oscillations, which is common with trigonometric functions like \(\sin(x)\), the integral can remain finite. For instance, setting \(g(x) = \sin(x)\) and using \(h(x) = g(x)\) allows us to satisfy all these requirements by ensuring the integral's oscillations are counterbalanced by the decreasing magnitude of \(h(x)\). Thus, by meeting these conditions, the integral \( \int_{a}^{\infty} g(x) \sin x \, dx \) is proven to converge as shown in the exercise solution!

Convergence of Integrals

The concept of convergence for integrals refers to whether an integral results in a finite value as its limits extend towards infinity. For improper integrals, convergence can often be non-intuitive, particularly when dealing with functions that oscillate or asymptotically approach values.
In our exercise, the focus is on the integrals of the form \( \int_{a}^{\infty} g(x) \sin x \, dx \) and \( \int_{a}^{\infty} g(x) |\sin x| \, dx \). Convergence of both integrals is heavily dependent on the behavior of \(g(x)\):

• When \(g(x)\) itself forms a convergent integral over the interval \( [a, \infty) \), this often implies a dampening of the oscillating \(\sin(x)\).
• If \(g(x)\) leads to a divergent integral, the oscillations in \(|\sin(x)|\) don't get "smoothed" out, resulting in an overall divergence.

Understanding the convergence properties of these integrals helps predict their outcomes without necessarily computing them through lengthy calculations. This is crucial for math students working with real-world data, where extrapolations can determine crucial trends or confirm models.

Continuous Functions

Continuous functions are foundational in calculus and analysis. For a function \(g(x)\) to be continuous on an interval \([a, +\infty)\), it must not have any breaks, jumps, or points of discontinuity across that interval. In simple terms, you can draw the graph of a continuous function without lifting your pencil from the paper.
In the given problem, one assumption is that \(g(x)\) is continuous and nonincreasing. This continuity ensures:

• Predictable behavior of \(g(x)\) across its entire domain.
• Facilitates integration since the boundaries of the integral (a to infinity) don't introduce complexities.
• The nonincreasing nature of \(g(x)\) combined with continuity implies it eventually approaches a limit, in this case, \(0\) as \(x\) heads towards infinity.

This property becomes critical in establishing convergence, as it ensures there won't be abrupt changes in \(g(x)\) that could otherwise disrupt the balancing act required by Dirichlet's Test. Therefore, continuity, along with the additional conditions laid out, acts as a guarantee for applying tests like Dirichlet's safely.