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Series

Series

At a Glance - The Integral Test

We already know that series and integrals share some similar properties. We're going to show that, by replacing the series with an equivalent integral, we can determine if the series converges.

In terms of our Pandora's box, we're just replacing our grilled cheese with something else. Assume we've replaced them with gremlins. Maybe we should keep the box shut on these guys.

Now, we're going to test the convergence or divergence of some series by doing some reasoning similar to what we did when studying left-hand and right-hand sums for integrals.

Integral Test: Let f be a non-negative decreasing function on [c, ∞) where c is an integer. If the integral

converges, then the series

converges.

If the integral diverges, then the series also diverges.

Like we mentioned before, the integral test replaces grilled cheese with gremlins. They're hard to tame, but if you can do so, you can open the box safely. If you can't tame the gremlins, then keep that box shut.

The integral test works because, depending on how we draw the series, we can choose whether the rectangles will cover more or less area than the integral.

In the example with the harmonic series we drew the series as an overestimate. Since the integral diverged, we knew the series had to diverge.

If we have an integral that converges, we draw the series as an underestimate (right-hand sum) instead. Since a convergent integral describes a finite area, the smaller area covered by the rectangles must also be finite.

Example 1

Does the constant series

where a > 0, converge or diverge?


Example 2

Does the harmonic series

converge or diverge?


Example 3

Does the series

converge or diverge?


Example 4

If possible, use the integral test to determine whether the series

converges or diverges.


Example 5

If possible, use the integral test to determine whether the series

Σ sin n

converges or diverges.


Exercise 1

If the terms of a series approach zero, must the series converge? Justify or provide a counterexample.


Exercise 2

Does

converge or diverge?


Exercise 3

For the series, determine if it's okay to use the integral test. If so, use the integral test to determine whether the series converges or diverges.


Exercise 4

For the series, determine if it's okay to use the integral test. If so, use the integral test to determine whether the series converges or diverges.


Exercise 5

For the series, determine if it's okay to use the integral test. If so, use the integral test to determine whether the series converges or diverges.


Exercise 6

For the series, determine if it's okay to use the integral test. If so, use the integral test to determine whether the series converges or diverges.


Exercise 7

For the series, determine if it's okay to use the integral test. If so, use the integral test to determine whether the series converges or diverges.


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