- 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

The first tool in our arsenal our convergence tests is for alternating series, which is a series whose terms alternate in sign. This is for a Pandora's box full of American and Swiss grilled cheese sandwiches. The alternating series test can tell us if it's safe to open that box.

The series

1 – 2 + 3 – 4 + 5 – 6 + ...

is an alternating series because the signs of the terms switch back and forth between positive and negative.

We can find a formula for the terms of an alternating series the same way we can find a formula for the terms of an alternating sequence.

Since the terms alternated sign the partial sums went up, then down, then up, like jumping back and forth over a line. Since the terms approached zero, the partial sums got closer and closer together, and closer and closer to whatever line they were jumping over.

Line jumping is the idea behind our first convergence test, the **alternating series test**. Since the terms of an alternating series change sign, the partial sums for any alternating series will jump back and forth over some line. If the terms are getting smaller and approaching zero, the partial sums will get closer to the line and so the series will converge.

**Alternating Series Test (AST):** If Σ *a _{n}* is an alternating series, and if

|*a _{n}*| > |

for all *n* (that is, the terms have strictly decreasing magnitude), and if

then the series converges.

The catch: We can't use the AST to conclude a series diverges. We can only use the AST to conclude that a series converges. If the AST doesn't tell us that the series converges, we need to use another test, which might include the divergence test.

Can we use the AST to conclude that the series

converges?

Answer.

We have an alternating series. For all *n*, we have

so the first condition is met. We also have

so the second condition is met. Since all conditions are met, the AST says that the series converges.

Suppose we know, now, that we have an alternating series Σ *a _{n}* that converges, and it converges to value

A partial sum *S _{n}* is just an approximation of

|*L* – *S _{n}*|.

This distance between the approximation *S _{n}* and the real sum of the series

In general, suppose we have a convergent alternating limburger series Σ *a _{i}* that sums to

We use the absolute value of the (*n* + 1)st term because the only thing the sign is good for is saying which way the jump goes.

To find the (*n* + 1)st term of the series we need to know the starting index of summation, or we won't know which term is the (*n* + 1)st.

Example 1

Can we use the AST to conclude that the series converges? |

Example 2

Can we use the AST to conclude that the series Σ(-1) converges? |

Example 3

Consider the alternating series Find an upper bound for the error in the 3rd partial sum. |

Exercise 1

Find the first 5 partial sums of the alternating series

What do you notice about the partial sums?

Exercise 2

Play with the alternating series

(a) Find the first 5 partial sums of this series. Write them so that each partial sum has a denominator of 32.

(b) Plot the sequence of partial sums on this graph:

(c) What do you notice about the behavior of the partial sums?

(d) Do you think this series converges? Why or why not?

Exercise 3

For the series, can we use the AST to say the series converges? If not, why not?

Exercise 4

For the series, can we use the AST to say the series converges? If not, why not?

Exercise 5

For the series, can we use the AST to say the series converges? If not, why not?

Exercise 6

For the series, can we use the AST to say the series converges? If not, why not?

Exercise 7

For the series, can we use the AST to say the series converges? If not, why not?

Exercise 8

Consider the alternating series

which converges by the AST. Find the error for each of the partial sums

(a) *S*_{9}

(b) *S*_{98}