The current proof that shows the Harmonic Series diverges is wrong. When you get to this stage:
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:
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:
Step two: Rewrite the Harmonic Series
With this, we can rewrite the Harmonic Series as
Step three: Find what is Missing
What is missing is all the numbers that can be written as , 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.