Convergence of Ln over N Series

Determine the convergence or divergence of the series $\sum_{n=1}^{\infty} \frac{\ln(n)}{n}$.

In tackling the problem of determining the convergence or divergence of the series given by the sum of the natural logarithm of n over n, we are delving into the realm of series and their convergence properties. This problem guides you through the intricacies of understanding series that do not have straightforward terms but instead involve logarithmic expressions.

One key strategy in approaching this problem is to consider the integral test, which is often powerful in assessing the convergence of series. By evaluating an improper integral from one to infinity, that closely resembles the terms of our series, we can ascertain the behavior of the series. This approach leverages your knowledge of integral calculus and helps in drawing parallel conclusions between integrals and corresponding infinite sums. Using the integral test in this context involves recognizing that the function involved is positive, continuous, and eventually decreasing, which are essential preconditions for applying this test correctly.

Moreover, problems of this kind also prompt consideration of other convergence tests, but the integral test remains highly illustrative here due to the logarithmic component involved. Exploring the behavior of the integral of the natural logarithm function allows students to see the convergence criteria in action and helps build an intuition towards series involving unbounded and nonlinear terms. Ultimately, understanding the nuances of logarithmic functions in series enables students to extend these methods to a wider class of problems involving power series or other transcendental functions.