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The current proof that shows the Harmonic Series diverges is wrong. When you get to this stage:

1 + 1/2 + (1/4 + 1/4) + ... = 1 + 1/2 + 1/2 + 1/2 + ...

You can always compare the terms, and if you compare them one-to-one, then you can say your new sum is larger. We want a real proof.

​What is the Harmonic Series?

The Harmonic Series is the sum of all the reciprocals of the positive integers. Or more commonly written out like this:

 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + ...

Step one: Create a set

The first step is to create a set. Our set would contain all the positive integers that cannot be written as an exponent of two other integers (not including one). Our set would look like this:

S = {2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, ...}

Step two: Rewrite the Harmonic Series

With this, we can rewrite the Harmonic Series as 1 + 1 + 1/2 + 1/4 + 1/5 + 1/6 + 1/9 + 1/10 + 1/11 + ...

Step three: Find what is Missing

What is missing is all the numbers that can be written as 1/k^n, where k, and n are integers. If we add all those fractions up, we get something less than one.

We have a contradiction. The extra one did too much to make up for the missing fractions. If this has a value, then mathematics would be broken. So therefore, the Harmonic Series diverges.

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