Use the Integral Test to determine the convergence or divergence of the series.
A)diverges
B)converges
C)Integral Test inconclusive

Question

Use the Integral Test to determine the convergence or divergence of the series. ​  ​
A)diverges 
B)converges 
C)Integral Test inconclusive

Quizplus · Accepted Answer

Consider the function f(x) = ln(x) / x. The function is defined for all x > 0 and is continuous and positive on the interval [1, ∞).

Now, for n ≥ 2, we have

f(n) = ln(n) / n ≤ ln(n) / 2

This is true because 2 ≤ n; since ln(x) is an increasing function, it follows that ln(n) ≥ ln(2) ≈ 0.69.

Using this inequality, we can estimate the nth term of the series:

a_n = (ln(n))^2 / n^2 
  = f(n) / n 
  ≤ (ln(n) / 2) / n 
  = ln(n) / (2n)

Integrating both sides from 1 to infinity, we get:

∫(1 to ∞) a_n dn ≤ ∫(1 to ∞) ln(x) / (2x) dx

The integral on the right-hand side can be evaluated using integration by parts:

u = ln(x), dv = dx / (2x) 
du = dx / x, v = (1/2) ln(x)

∫(1 to ∞) ln(x) / (2x) dx 
= [(1/2) ln(x)^2]_(1 to ∞) - ∫(1 to ∞) dx / (2x) ln(x) 
= (1/2) lim_(n→∞) ln(n)^2 - [(1/2) ln(x)]_(1 to ∞) 
= ∞

Since the improper integral diverges, it follows that the series ∑ a_n also diverges by the Integral Test. Therefore, the answer is A) diverges.

Use the Integral Test to Determine the Convergence or Divergence