An integral expression for n!

I gave a challenge question at the end of class a week or so ago. Here I will give the solution and show that it gives us something rather strange and surprisingly useful.

I wrote down the following, and asked you to prove it:

$\int_0^\infty e^{-t} t^N dt=N!$

For $N\ge 0, N\in \mathbb{Z}$. Now, N! can be thought of as the number of different orderings of pulling N objects out of a bag (without replacement) when they are all different. If you have N things in a bag, then there are N possible things that you can pull out first. There are then N-1 ways of pulling out the next object, N-2 ways of pulling out the next, etc. and finally, when you’ve pulled out N-1 objects there’s only a single possibility of pulling out the last. So:

$N!=N(N-1)(N-2)(N-3)...3.2.1$

And the number of ways of pulling no objects out of a bag is 1, because you just don’t pull anything out.…

A general formula for partial fractions

Out of the blue I wrote down a rather confusing mass of indices and summations on the board a few days ago. Writing this down at the last minute was perhaps a bad idea, but I wanted to show what the general form for expanding a fraction into partial fractions was. Here I’m just motivating it a little more. It’s not something that you will need to use, but it’s often good to write things down in as general a form as you can.

Let’s say that we have an expression of the form:

$\frac{P(x)}{(x+3)(x-2)^2}$

Where P(x) is some polynomial of degree less than 3 (because the denominator is degree 3). We can write this as:

$\frac{A}{x+3}+\frac{B}{(x-2)^2}+\frac{C}{x-2}$

To find A,  B and C, you cross-multiply, and then match coefficients of powers of x with those in P(x).  If you have an irreducible quadratic in the denominator you will have terms of the form:

$\frac{Ax+B}{ax^2+bx+c}$

in your partial fraction, and of course if it’s an irreducible quadratic to an integer power greater than one, you will have multiple terms, just as you do for the (x-2) expression in the example above.…

The 17 equations that changed the world

I was so excited the first time I read Ian Stewart’s book entitled “The 17 equations that changed the world“. The book is written in simple and easy to understand language with interesting practical examples for applications. I immediately wrote to Ian Stewart requesting if I could reproduce his work in form of posters in both English and French to be used at AIMS-IMAGINARY in Senegal in 2015 (see here). I remember his only reply was “please proceed but I won’t be able to attend since I have prior committments”. Business Insider published a list of these equations emphasing further how intuitive they are (see here). I do strongly believe every school in the world be it elementary, college, secondary, technical, university, you name it, should have these posted up or painted on the walls of their science departments/offices, classrooms, laboratories etc; in all langauges applicable.…

Where did that substitution come from?

If you want to understand maths, you really have to do it. I recommend going through these examples and using the substitutions given here as hints. Get a blank piece of paper, put your notes away and try to do these examples and see if you get the same answers as in class. If you don’t, write in the comments, and we can see where things may have gone astray.

I’ve been teaching integration by substitution, including by trig substitutions over the last few days, and a frequent question which a newbie substituter will ask is “how did you know to make that substitution?”. It’s a very reasonable question, and one that takes practice to build the correct intuition, but I’ll do my best to give some motivation now as to why we made some of the substitutions we made. We won’t solve the integrals, but we will motivate here why we make particular choices for substitutions.…