In the Real World
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.