Question

Let \(f_{n}(x)=a e^{-n a x}-b e^{-n b x}\) where \(0

Short answer

a. Diverges; b. Converges to 0; c. Result is \(\log(b/a)\).

Step-by-step solution

Explore the Integral of Absolute Value

To solve part (a), calculate the integral \( \int_{0}^{\infty} \left| f_{n}(x) \right| dx \). As \( \int_{0}^{\infty} e^{-c x} dx = \frac{1}{c} \) for \( c > 0 \), consider both terms separately. Under its absolute sign, compare the behavior of terms as \( n \to \infty \).

Calculate Integral \( \int_{0}^{\infty} |f_n(x)| dx \)

Since \( b > a \), \( e^{-nbx} \) decreases faster than \( e^{-nax} \). Therefore, for large \( n \), \( |f_n(x)| \approx |a e^{-nax} - b e^{-nbx}| \) reduces to \( a e^{-anx} - b e^{-bnx} \). Evaluate this integral to show divergence: \( \int_{0}^{\infty} |f_n(x)| dx \to \infty \) as both components involve substantively large domains for \( n \).

Evaluate Convergence of Series of Integrals

To prove \( \sum_{1}^{\infty} \int_{0}^{\infty} |f_{n}(x)| dx = \infty \), apply comparison tests, examining series such as \( \sum_{1}^{\infty} \frac{1}{an} \), showing each integral \( \int_{0}^{\infty} |f_n(x)| dx \to \infty \) as n-dependence behaves in divergent manner.

Calculate Integral of Function

To solve part (b), calculate \( \int_{0}^{\infty} f_{n}(x) dx \) directly:\[ \int_{0}^{\infty} f_{n}(x) dx = a \int_{0}^{\infty} e^{-nax} dx - b \int_{0}^{\infty} e^{-nbx} dx = \frac{a}{na} - \frac{b}{nb} \]This simplifies to \( \frac{1}{n} - \frac{1}{n} = 0 \).

Evaluate the Summation Using Integral Result

For part (b), observe the result \( \int_{0}^{\infty} f_{n}(x) dx = 0 \). Therefore, the series \( \sum_{1}^{\infty} \int_{0}^{\infty} f_{n}(x) dx = 0 \), as each term is zero.

Apply Fubini's Theorem for Interchange of Series and Integral

For part (c), use Fubini's theorem, which justifies the interchange because the sum converges in \( L^1(0, \infty) \).Given the result \( \int_{0}^{\infty} f_{n}(x) dx = 0 \), thus \( \sum_{1}^{\infty} f_{n} \in L^1 \).

Calculate Finally Intertwined Sum and Integral

Compute:\( \int_{0}^{\infty} \sum_{1}^{\infty} f_{n}(x) dx \). Use explicit series results interchangeably via Fubini's theorem:Given evaluations of exponential terms, find the result \( \log(b/a) \).

Key concepts

Fubini's Theorem

Fubini's Theorem provides a powerful tool that allows us to interchange the order of integration and summation in many complex mathematical problems. This is particularly useful when evaluating integrals of series of functions.
For it to apply, we need to be certain of two conditions:

• The integral of the absolute value of the function is finite, which means it needs to be measurable and integrable.
• The series of the function converges absolutely, a requirement for moving the sum outside the integral.

In our exercise, Fubini's theorem is crucial. It helps us understand how we can switch the order of \[\int_0^\infty \sum_{1}^{\infty} f_{n}(x) \, dx = \sum_{1}^{\infty} \int_0^\infty f_{n}(x) \, dx.\]Due to the properties of the given function sequence and convergence, we arrive at a finite result, \( \log(b/a) \), a testament to the theorem's efficiency.

Convergence of Series

The Convergence of Series is a fundamental concept to determine whether an infinite series sums up to a finite value or not.
It’s essentially the idea of adding up an infinite number of terms such that the total is a definite number.
For part (a) and (b) of the exercise, understanding the convergence is key.

• Part (a) demonstrates a series whose terms grow unbounded, making it divergent.
• In contrast, part (b) shows a harmonic balance in the integral terms, leading to convergence at zero.

The concept includes testing conditions such as absolute and conditional convergence. It allows defining a series' behavior. Tests like comparison, ratio, and root tests are often employed to judge a series' convergence effectively. When a series converges, Fubini's theorem can be plausibly applied, as seen in part (c) of the exercise, making these concepts inevitably connected.

Exponential Functions

Exponential Functions are a class of functions where the variable appears as an exponent. They have distinctive properties that are of interest in many analyses.
These functions are typically written as \( f(x) = a e^{bx} \), where \( e \) is the base of the natural logarithm and is approximately equal to 2.718.
In our exercise, the exponential function embodies decaying terms like \( e^{-na x} \) and \( e^{-nb x} \).

• The function's form makes it central to calculating integrals as it dictates behaviors such as rapid decrease or decay over the domain.
• Their derivatives and integrals exhibit unique traits; notably for \( e^{cx} \), its integral from zero to infinity is \( \frac{1}{c} \) if \( c > 0 \), ensuring convergence to a finite value.

Understanding these properties is essential for analyzing the infinite nature of the series in the given exercise, determining where convergence occurs, and applying the results accordingly.