- 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've seen that geometric series can get used to calculate how much money you've got in the bank. They can also be used to calculate the amount of medicine in a person's body, if you know the dosing schedule and amount and how quickly the drug decays in the body.

On a more fun note, the harmonic series is a divergent infinite series. There's also a harmonic series in music, and they're very closely related.

When you pluck a string on a musical instrument, it creates more than one note. It creates a *base note* and also a collection of higher notes called *harmonics* or the (musical) *harmonic series*.

The different musical harmonics correspond to the different terms of the mathematical harmonic series.

When a string is plucked, it vibrates along its whole length to form the base note.

At the same time, it vibrates in two pieces to form the first harmonic of the musical harmonic series.

It vibrates in three pieces to form the next harmonic of the musical harmonic series.

And so on, and so on.

If we say the original length of the string is 1, then the base note and harmonics of the musical harmonic series correspond to the lengths

These are the terms of the mathematical harmonic series.

This video demonstrate the musical harmonics inside a piano.

This page has an interactive demo that lets you see how the string vibrates and hear the corresponding musical harmonics at the same time.