Recall that the following inequalities, based on the integral test, give the lower and upper bounds for the approximation of the total sum:

     

For the given series Σ(1/n³) with n = 5, the inequlities become

     

The partial sum can be determined as follows:

     

The improper integral can be evaluated as:

The inequalities become

Taking the midpoint as the total sum, hence

     

with

     Error = (1.2057 - 1.1996)/2 = 3.05 x 10^-3