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

EXAMPLE 2:  Determine whether the series

converges or diverges.

For large  the dominant term in the denominator is  , so we compare the

given series with the series  . Observe that

because the left side has a bigger denominator. (In the notation of the Comparison Test,  is the left side and is the right side.) We know that

is convergent because it’s a constant times a p-series with . Therefore

is convergent by the Comparison Test.

Note  Although the condition  in the Comparison Test is given for all   , we need verify only that it holds for  , where  is some fixed integer, because the convergence of a series is not affected by a finite number of terms. This is illustrated in the next example.

EXAMPLE 3:  Test the series

for convergence or divergence.

We can test it by comparing it with the harmonic series. Observe that  and so   

We know that   is divergent . Thus the given series is divergent by the Comparison Test.