* Example.* Does the series converge or diverge?

is a p-series with , so it converges. is a geometric series with , so it diverges.

Hence, the sum of the two series *diverges*.

* Example.* Determine whether the series
converges or diverges. If it converges, find its sum.

The series is geometric with ratio , so it converges. The sum is

* Example.* Does the series converge or diverge?

The series is geometric with ratio . Therefore, the series diverges.

* Example.* Does the series converge or diverge?

The series diverges by the Zero Limit Test.

* Example.* Does the series
converge or diverge?

is a geometric *sequence* with ratio
, so as . Therefore,

Hence, the series diverges, by the Zero Limit Test.

* Example.* Does the series converge
or diverge?

The terms are positive. The function is continuous for . The derivative is

which is clearly negative for . Thus, the terms of the series decrease. The hypotheses of the Integral Test are satisfied.

Compute the integral:

The integral converges, so the series converges, by the Integral Test.

* Example.* Does the series converge
or diverge?

Notice that as ,
*decreases* to 1. Thus,

is harmonic, so it diverges. Therefore, the original series diverges by direct comparison.

* Example.* Does the series converge or diverge?

For large n, . Do a Limit Comparison:

The limit is a finite positive number. diverges, because it's a p-series with . Therefore, the original series diverges, by Limit Comparison.

* Example.* Does the series
converge or diverge?

Apply the Ratio Test:

Therefore, the series converges by the Ratio Test.

* Example.* Does the series converge or diverge?

Apply the Root Test:

The series converges by the Root Test.

* Example.* The series converges by the Alternating Series Test.

Find the smallest value of n for which the partial sum approximates the actual value of the sum to within 0.01.

The error in using to approximate the actual value of the sum is less than the term in absolute value, so I want

The smallest value of n is .

* Example.* Does the series
converge absolutely, converge conditionally, or diverge?

Consider the absolute value series . By Limit Comparison,

The limit is a finite positive number. diverges, because it's harmonic. Therefore, diverges by Limit Comparison.

Hence, does not converge absolutely.

Consider the original series . It alternates, and if ,

Hence, the terms decrease in absolute value. Finally,

By the Alternating Series Test, converges. Since it converges, but does not converge absolutely, it converges conditionally.

Copyright 2016 by Bruce Ikenaga