In the first part of these notes we showed the following \[ S(n) = \sum_{k = 1}^{k = n}\frac{f(k)}{k!} = \prod_{k = 2}^{k = n} (1 + \frac{f(k)}{a_k}) = P(n)\] for all \(n > 1\) where the sequence \((a_k)\) is given by the recurrence relation \( a_2 = 2\) and \(a_{k+1} = (k+1)(a_k + f(k))\).

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 \(S(n) = \sum_{k = 1}^{k = n}\frac{1}{k!} \). So \( \prod_{k = 2}^{k \rightarrow \infty} (1 + \frac{1}{a_k}) = e-1 \) where \( a_2 = 2 \) and \(a_{k+1} = (k+1)(a_k +1)\) by considering Taylor expansion of \(e\).

Here is another example : we can write the harmonic series as \(\sum_{n=1}^{n \rightarrow \infty} \frac{1}{n} \). So \( \prod_{k = 2}^{k \rightarrow \infty} (1 + \frac{(k-1)!}{a_k}) \) where \( a_2 = 2 \) and \(a_{k+1} = (k+1)(a_k +(k-1)!)\)

**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 \( \prod_{k = 2}^{k \rightarrow \infty} (1 + \frac{(k-1)!}{a_k}) \). Can we prove that this diverges without assuming the summation series \(\sum_{n=1}^{n \rightarrow \infty} \frac{1}{n} \)?

**2)** Can we re-write the sequence \(a_2=2\) and \(a_{k+1} = (k+1)(a_k +f(k))\) 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!

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

There are no comments in this discussion.