Here's something interesting I found out last year. I don't know if its been done before but I strongly suspect it has.
Consider a summation series of form such that .
Now let's do something unusual and create the following sequence starting with the term and then for all .
Finally we'll focus our attention on the following product series for all .
We'll show that for all .
Proof : By induction on n.
We'll start with the base case.
We have as required.
This part is a quite long, so we'll break it down into 3 parts.
Part 1 : Re-arranging the Factors of the Product Series
We can write each factor from our product series as by using the definition of our sequence.
Part 2 : Re-writing our Product Series
Now we have that
Part 3 : Final Step
Suppose for some natural number .
Then as required.
End of Proof
We'll go through some of the consequences of this in the next note!