The series is .

.

.

Adding and subtracting a finite number of terms from a series do not affect the convergence or

divergence of the series.

Thus, if is convergent then is also convergent.

Consider .

The series is a -series with .

-series test:

The -series is convergent if and divergent .

Thus the series is divergent.

Compare the series with .

.

The Comparison Test:

Suppose that and  are series with positive terms.

(i) If  is convergent and  for all , then  is also convergent.

(ii) If  is divergent and  for all , then is also divergent.

Thus, series greter than the diverging series is also divergent.

  is divergent  by part (ii) of the comparison Test.

Therefore, is also divergent.

is also  divergent.