It's multiplied by the alternating harmonic series. Then simplify the stuff in parentheses to getДЇactor out from each term (that is, factor out 2 from each denominator). We can write the terms of the alternating harmonic series like this:ДЇirst take a moment to convince yourself that all the terms of the alternating harmonic series do show up in this rearrangement.

The alternating harmonic series is conditionally convergent, and when we get to Taylor series we'll see that it sums to ln 2. The alternating harmonic series is a good example of this weirdness. In fact, the terms can be rearranged to add up to anything you want. But if a series converges conditionally, rearranging the terms can make them add up to different values. If a series converges absolutely you can still rearrange the terms however you want. When you're adding infinitely many numbers, the rules change. that's true when you're adding finitely many numbers. No matter what order you add the numbers, you'll always get the same answer. You've known for a long time that addition is commutative. I Like Abstract Stuff Why Should I Care?.This page has an interactive demo that lets you see how the string vibrates and hear the corresponding musical harmonics at the same time. This video demonstrate the musical harmonics inside a piano. These are the terms of the mathematical harmonic series. 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 It vibrates in three pieces to form the next harmonic of the musical harmonic series. When a string is plucked, it vibrates along its whole length to form the base note.ДЄt the same time, it vibrates in two pieces to form the first harmonic of the musical harmonic series. The different musical harmonics correspond to the different terms of the mathematical harmonic series. It creates a base note and also a collection of higher notes called harmonics or the (musical) harmonic series. When you pluck a string on a musical instrument, it creates more than one note. There's also a harmonic series in music, and they're very closely related. On a more fun note, the harmonic series is a divergent infinite series. 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. We've seen that geometric series can get used to calculate how much money you've got in the bank.

Harmonic series calculus