integral test on a series of numbers.

Question

Prove that you can use the integral test on the series which is the sum from n=1 to infinity of the

terms a_n= ln(n)/n.

Then use the test to establish the convergence or divergence of the series

Answer 1

Let

f(x) = ln(x)/x

The integral test requires that on the interval [1, ∞), f(x) be

defined and

monotone decreasing.

That is it defined should be obvious.

To see if it is monotone decreasing, we see if it's derivative is negative.

f'(x) = ((1/x)x - ln(x))/x² = (1 - ln(x))/x².

So the derivative is negative for x > e, not for x ≥ 1.

But that does not really matter, because

the given sum from 1 to ∞ converges if and only if the sum from 3 to ∞ converges.

Therefore it is sufficient to apply the test on the interval [3, ∞).

So we can conclude that the sum from 3 to ∞ converges iff

the integral from 3 to ∞ ∫f(x)dx is finite.

Let F(x) = ∫ln(x)/x dx = ln²(x)/2

Then the definite integral equals F(∞) - F(3), which is not a real number.

Therefore the given series diverges.