Question

Use the Integral Test to determine the convergence or divergence of the series. $$ \frac{\ln 2}{2}+\frac{\ln 3}{3}+\frac{\ln 4}{4}+\frac{\ln 5}{5}+\frac{\ln 6}{6}+\cdots $$

Short answer

By the Integral Test, the given series is divergent since the corresponding improper integral is divergent.

Step-by-step solution

Verification of Conditions

Verify the conditions for applying the Integral Test. The three conditions are that the function should be positive, decreasing and continuous on the interval [2, infinity). The function f(x) = ln(x)/x meets all these conditions.

Evaluation of the Integral

Evaluate the integral from 2 to infinity of (ln(x)/x) dx. This is done by applying the rule of integration by parts, where u = ln(x) and dv = 1/x dx. This integral computation will give a result that indicates the behaviour of the integral.

Convergence/Divergence of the Series

Declare the series to be convergent or divergent based on the behaviour of the integral. If the integral is convergent, the series is also convergent. If the integral is divergent, the series is also divergent.

Key concepts

Convergence of Series

When learning about infinite series, one of the fundamental concepts is understanding whether a series converges or diverges. A series is a sum of a sequence of terms, and if these terms approach a specific value as you keep adding more terms, the series is said to converge. On the other hand, if the sum grows without bound or fails to approach a specific limit, we describe the series as diverging.

To ensure the right approach to determining convergence, several tests can be applied. One such test is the Integral Test. It involves taking a related improper integral and establishing whether it converges or diverges. If the integral of a function, from some point to infinity, has a finite value, then the corresponding series also converges. Conversely, if the integral diverges, so does the series. It is important to check the prerequisites of the test before applying it: the series' terms must correspond to a function that is continuous, positive, and decreasing.

Integration by Parts

A powerful technique for evaluating integrals that cannot be computed using standard methods is integration by parts. It is derived from the product rule for differentiation and provides a way to integrate products of two functions.

The formula is given by: \[\int u dv = uv - \int v du\] where \( u \) and \( dv \) are parts of the integrand chosen by the user. Selecting these parts strategically is a skill developed with practice, as a poor choice could complicate the integral further. When using integration by parts for the Integral Test, a careful selection helps in simplifying the improper integral associated with the test.

Infinite Series

An infinite series is essentially a sum of infinitely many terms, expressed as a sequence. The terms may arise from a function or a series of constants and may involve some pattern or rule. Infinity is a concept that challenges our straightforward notions of sum, because it's not about adding 'a lot' of terms—rather, it's about a process that never ends.

Understanding whether such series converge or diverge is crucial in many fields of mathematics and science. Depending on the terms, an infinite series might approach a finite limit or continue to grow indefinitely. The series evaluated using the Integral Test is a particular type of infinite series where each term is represented by a function over a sequence of integers, allowing for assessment of convergence.

Improper Integrals

An integral is deemed improper if one of the limits of integration is infinity, or if the integrand becomes infinite within the limits of integration. In the latter case, this usually happens at a point where the function has a discontinuity or vertical asymptote.

To evaluate an improper integral, it is often transformed into a limit problem where a definite integral is taken over increasing intervals that approach the point of discontinuity or infinity. The Integral Test for convergence of series relies on determining whether a related improper integral converges. Thus, understanding improper integrals is essential when we're applying the Integral Test to an infinite series, as they will indicate the behavior of the sum in question.