- Topics At a Glance
- Series: This is the Sum That Doesn't End
- Sigma Notation
- Alternating Series
- Convergence of Series
- Finally, Meaning...and Food
- Relationship Between Sequences
- Math-e-magics?
- When Limits of Summation Don't Matter
- Properties of Series
- Special Cases
- Arithmetic Series
- Geometric Series
- Finite Geometric Series
- Infinite Geometric Series
- Decimal Expansion
- Word Problems
- Visualization of Series
- When Limits of Summation Don't Matter
**Tests for Convergence**- The Divergence Test
- The Alternating Series Test
- The Ratio Test
**The Integral Test**- The Comparison Test
- Absolute Convergence vs. Conditional Convergence
- Summary of Tests
- Taylor and Maclaurin Series
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

We already know that series and integrals share some similar properties. We are 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 are 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.

Before, we used graphs the convergence or divergence of some series and 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 are 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 |

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 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

Exercise 7