Question

Use the direct comparison Test to determine the convergence or divergence of the series.

Answer 1

Step 1:

The series is .

Direct comparison test:

Let for all .

1.If convergence, then convergence.

2.If diverges, then diverges.

The dominant part of the numerator is and the dominant part of the denominator is .

Now compare the given series with the series .

Observe that .

Because the numerators are equal and denominators are 1 grater in .

Step 2:

The obtained series is .

The series is in the form of geometric series .

In this case and .

is geometric series.

Convergence of a geometric series:

A geometric series with common ratio diverges if .If then the series converges to the sum .

with ratio .

The series is converges to the sum of series.

The series is converges to .

Step 3:

Direct comparison test:

If convergence, then convergence.

If the series is converges, then is converges.

Solution:

The series is converges.