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Problem : Use the integral test to show that 1/(n(log(n))2) converges.

Letting u = log(x), we compute the relevant integral:


dx=du  
 =|log(2)log(m)  
 = -  

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:


dx=  
 =du  
 =(log(u)|2m3+1)  
 =log  

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.
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