Problem : Use the integral test to show that 1/(n(log(n))2) converges.Letting u = log(x), we compute the relevant integral:
The limit of this integral as m→∞ exists and is equal to 1/log(2), so the sum converges by the integral test.
Problem : Show that the series
diverges using the integral test. Letting u = x3 + 1, we compute the relevant integral:
This quantity clearly has no limit as m→∞, so the series diverges. Notice that there are far more efficient ways to show the divergence of this series; for instance one could use the comparison test with the harmonic series 1/n.