- 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 know all we need to know about geometric series. As a nifty bonus, we can use geometric series to better understand infinite repeating decimals. This is something you can rub in your former math teachers' faces.

Does 0.99999... equal 1?

Answer.

First we have to figure out what 0.99999... means. If we break it up by individual decimal places,

0.99999... = 0.9 + 0.09 + 0.009 + 0.0009 + 0 .00009 + ....

Convert each term to a fraction. Now

This is an infinite geometric series with and . The sum of the series is

We've just proved that

.99999... = 1.

Haha! Take that 8th-grade math teacher that took off 5 points from your test when you wrote '1' instead of '0.9999...'.

The same trick can be used to turn other infinite repeating decimals back into rational numbers.

Exercise 1

What is

0.777...

when written as a fraction?

Exercise 2

Write

0.787878...

as a rational number.