In the first part of these notes we showed the following for all where the sequence is given by the recurrence relation and .
In this note we'll have a look at some applications of this.
Part 1 : A couple of Examples
For this constant function we have . So where and by considering Taylor expansion of .
Here is another example : we can write the harmonic series as . So where and
Part 2 : Questions and Remarks for Brilliant Here's some questions I have for the Brilliant Community.
1) We showed that we can re-write the harmonic series as . Can we prove that this diverges without assuming the summation series ?
2) Can we re-write the sequence and with out recursion? Do these recursive sequences have any combinotorical properties?
3) We haven't really learnt anything significantly new from re-writing these series and in product series form this is computationally difficult due to the recursive sequences.
4) Post any interesting examples you can re-write below!