Problem 27 - Calculus: One and Several Variables

Question

Determine whether the series converges or diverges. $$\sum \frac{\ln k}{k^{2}}$$

Step-by-Step Solution

Answer

The series \(\sum_{k=1}^\infty \frac{\ln k}{k^2}\) converges.

Step 1: Identify the series and function

The given series, known as an infinite sum, is represented as \(\sum_{k=1}^\infty \frac{\ln k}{k^2}\). The corresponding function would be \(f(x) = \frac{\ln x}{x^2}\)

Step 2: Apply the Integral Test

We will integrate the function f(x) from 1 to infinity to find if the series converges or diverges. \(\int_1^\infty \frac{\ln x}{x^2}dx\)

Step 3: Perform integration by substitution

To find the integral of \(\frac{\ln x}{x^2}\), we can use integration by substitution. Let \(u = \ln x \quad \Rightarrow \quad du = \frac{1}{x}dx\) and \(v = -\frac{1}{x} \quad \Rightarrow \quad dv = \frac{1}{x^2}dx\) Now rewrite the integral using u and v, and the limits of integration accordingly: \(\int_1^\infty \frac{\ln x}{x^2}dx = -\int_0^{\infty} u e^{-u} du\)

Step 4: Apply Integration by Parts

Integration by parts is defined as \(\int u dv = uv - \int v du\). Let's use this to solve the integral from Step 3. \(I = -\int_0^\infty u e^{-u} du\) Let \(p = u \quad \Rightarrow \quad dp = du\) and \(q' = e^{-u} \quad \Rightarrow \quad q = -e^{-u}\) Applying the integration by parts formula: \(I = - (pq\big|_0^\infty - \int_0^\infty q dp)\) Evaluate the integral: \(I = - (-u e^{-u} \big|_0^\infty + \int_0^\infty e^{-u} du) = -(0 - \int_0^\infty e^{-u} du)\) Now evaluate the remaining integral: \(I = \int_0^\infty e^{-u} du = -e^{-u}\big|_0^\infty = (-0) - (-1) = 1\) Since the integral \(\int_1^\infty \frac{\ln x}{x^2}dx\) converges, then the series \(\sum_{k=1}^\infty \frac{\ln k}{k^2}\) also converges by the Integral Test.

Step 5: Conclusion

The series \(\sum_{k=1}^\infty \frac{\ln k}{k^2}\) converges.

Key Concepts

Integral Test

The Integral Test is a handy tool in calculus for determining the convergence or divergence of an infinite series. This test applies to series of the form \(\sum_{n=1}^\infty a_n\) where \(a_n = f(n)\) and \(f(x)\) is a continuous, positive, decreasing function on the interval \([1, \infty)\).
For the Integral Test:

• Calculate \(\int_1^\infty f(x)\,dx\).
• If the integral converges (i.e., results in a finite value), then the series converges.
• If the integral diverges (i.e., it is infinite), then the series diverges too.

The beauty of the Integral Test lies in its simplicity. By translating a series problem into an integral problem, it can sometimes be easier to determine convergence or divergence. Remember, this test requires that the function \(f(x)\) meet all the conditions mentioned for an accurate result.

Integration by Parts

Integration by Parts provides a method for integrating the products of functions. It comes in handy especially when faced with integrals involving products where substitution alone won't do the trick. The formula for integration by parts is:

• \(\int u \ dv = uv - \int v \ du\).

Here, \(u\) and \(dv\) are parts of the original integral; \(du\) and \(v\) are their respective derivatives or integrals.To decide on \(u\) and \(dv\), remember the acronym "LIATE" (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential), which helps in selecting which function to differentiate (take as \(u\)) and which to integrate (take as \(dv\)). In our series problem, separation into \(u = \ln x\) and \(dv = \frac{1}{x^2} dx\) allowed for simplification and ultimately led to convergence evaluation.

Infinite Series

An infinite series is a sum of infinitely many numbers, or terms. This particular concept occurs frequently in calculus and applies in various mathematics and sciences. Infinite series are usually written using summation notation, which looks like this: \(\sum_{k=1}^\infty a_k\).There are two main categories:

• Convergent Series: The sum of the series approaches a finite limit.
• Divergent Series: The sum of the series does not approach a finite limit.

The convergence of a series, like \(\sum \frac{\ln k}{k^2}\), is crucial in calculus since it determines the ability to approximate or calculate infinite processes. Calculus offers tests, like the Integral Test, to determine whether these series converge or diverge.

Calculus

Calculus is a branch of mathematics built from the study of change, employing techniques to understand functions, limits, derivatives, integrals, and infinite series. Within calculus, various tools help to solve complex mathematical problems:

• Derivatives measure how functions change and are often involved in rate-related problems.
• Integrals represent the accumulation of quantities and the calculation of areas under curves.
• Infinite Series provide a framework for summing sequences indefinitely, offering solutions to real-world problems in physics and engineering.

Calculus introduces powerful methods for exploring and understanding the behavior of continuous functions. With it, mathematicians and scientists can better predict and analyze physical systems, economics, and even biological processes.