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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 = x 3 + 1 , we compute the relevant integral:

dx =  
  = du  
  = (log(u)|2 m3+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 .