November 2020 – Mathemafrica

A challenging limit

This post comes mostly from the youtube video by BlackPenRedPen found here: https://www.youtube.com/watch?v=89d5f8WUf1Y&t=3s

This in turn comes from Brilliant.com – details and links can be found in the original video

In this post we will have a look at a complicated-looking limit that has an interesting solution. Here it is:

$\lim_{n \rightarrow \infty} ( \frac{n!}{n^n})^{\frac{1}{n}}$

This looks pretty daunting – but we will break the solution down into sections:

taking the logarithms and rearranging
recognising something familiar
finding the numerical value

Step 1: Taking the Logarithm

The first step here is to take the logarithm, a generally useful trick when applying limits. First we assign the variable L to the limit (so that we can solve for it in the end). Now lets do some algebra:

$L = \lim_{n \rightarrow \infty} ( \frac{n!}{n^n})^{\frac{1}{n}}$

$\ln(L) = \ln(\lim_{n \rightarrow \infty} ( \frac{n!}{n^n})^{\frac{1}{n}})$

Noting that the natural logarithm $\ln$ is a continuous function and therefore we can take the limit outside of the function:

$\ln(L) = \lim_{n \rightarrow \infty} \ln( (\frac{n!}{n^n})^{\frac{1}{n}})$

Next we can use the logarithm laws to bring down the exponent:

$\ln(L) = \lim_{n \rightarrow \infty} \frac{1}{n} \ln(\frac{n!}{n^n})$

Alright, now we have taken the logarithm, step 1 is complete.…

Parrondos Paradox

Introduction

In this post we will have a look at Parrondos paradox. In a paper* entitled “Information Entropy and Parrondo’s Discrete-Time Ratchet”** the authors demonstrate a situation where, by switching between 2 losing strategies, we can create a winning strategy.

Setup

The setup to this paradox is as follows:

We have 2 games that we can play – if we win we get 1 unit of wealth, if we lose, it costs 1 unit of wealth. Game A gives us a payout of 1 with a probability of slightly less than 0.5. Clearly if we play this game for long enough we will end up losing.

Game B is a little more complicated in that it is defined with reference to our existing winnings. If our current level of wealth is a multiple of M we play a game where the probability of winning is slightly less than 0.1. If it is not a multiple of M, the probability of winning is slightly less than 0.75.…