Question

Examine the convergence of the following series using the integral test:

a) ∑[n=1 to [infinity]] ln(n) / n

Answer 1

The integral test compares a series to an improper integral to determine convergence.

For the series ∑[n=1 to ∞] ln(n) / n, let f(x) = ln(x) / x. To examine convergence, evaluate the integral ∫[1 to ∞] ln(x) / x dx. This integral can be challenging to solve analytically.

Step-by-step explanation:

The integral test states that if f(x) is continuous, positive, and decreasing for x ≥ 1, and if ∫[1 to ∞] f(x) dx converges, then the series ∑[n=1 to ∞] f(n) also converges.

In this case, as ln(x) / x is a decreasing function for x ≥ 1, the integral test applies. However, the integral ∫[1 to ∞] ln(x) / x dx cannot be easily integrated using elementary functions.

To evaluate the convergence of the series, consider the behavior of ln(n) / n as n approaches infinity. Although the function decreases as n increases, it does not decrease fast enough for the integral to converge. The natural logarithm function grows slowly, causing the series to diverge by comparison to the harmonic series, which is known to diverge.

Therefore, based on the behavior of ln(n) / n as n approaches infinity and the comparison with the harmonic series, the series ∑[n=1 to ∞] ln(n) / n also diverges.