Method of Differences

[Edit: Oh my goodness, I took a look at Professor Adrian Yeo's derivation, and it's nearly the same! Or rather, equivalent! I feel good! Haha.]

I was doing some ‘A’-Levels tuition when I chanced upon a method to determine the general formula for many types of series. In fact, I'm about to derive the sum to n terms for the following series:


To start, let’s try this method out on the harmonic series:


And of course we can simplify this into:


So how do we start? Well, personally, I like to start with the Method of Differences, which is a very useful methodology taught now in ‘A’-Levels (but wasn’t taught in my time!), but often under-rated method. So let’s find the difference between:


This being the case, we then carry out a summation to n terms:



But do think about it:


And we already know that:


And therefore we write:

With this, we see that:


And immediately, we can substitute this back into equation (a):


Now the ingenuity of this method is that it allows us to determine the sum to n terms of the following series:


And voila, the sum to infinite terms can be seen to be a finite sum:


What an excellent and neat proof! :)