Infinite Limits in Python

This talk got re-worked as The $\sin(x)$ Button.

This talk uses a bit of Python code.

You can download the relevant code here.

First, we consider the following mysterious function in Python.

def mystery(x, N):
    """This mystery function outputs some value based on x and N.
    We assume that x is a real number, and N is a positive integer."""

    output = 0;

    for n in range(0, N + 1):
        output += ((-1)**n)*(x**(2*n+1))/(math.factorial(2*n+1));

    return output;

Our big question today is:

What is this function calculating?

We will try various strategies to get a handle on it.

Calculate some specific values of the function.
Write a function table(C,R) which produces a similar table of values with columns $x = 0, 1, 2, \dots, C$ and rows $N = 1, 2, \dots, R.$ Your function should return a two dimensional array of values.

For reference, the output of table(10,10).

	$x = 0$	$x = 1$	$x = 2$	$x = 3$	$x = 4$	$x = 5$	$x = 6$	$x = 7$	$x = 8$	$x = 9$	$x = 10$
$N = 1$	$0.0000$	$0.8333$	$0.6667$	$-1.5000$	$-6.6667$	$-15.8333$	$-30.0000$	$-50.1667$	$-77.3333$	$-112.5000$	$-156.6667$
$N = 2$	$0.0000$	$0.8417$	$0.9333$	$0.5250$	$1.8667$	$10.2083$	$34.8000$	$89.8917$	$195.7333$	$379.5750$	$676.6667$
$N = 3$	$0.0000$	$0.8415$	$0.9079$	$0.0911$	$-1.3841$	$-5.2927$	$-20.7429$	$-73.5097$	$-220.3683$	$-569.4268$	$-1307.4603$
$N = 4$	$0.0000$	$0.8415$	$0.9093$	$0.1453$	$-0.6617$	$0.0896$	$7.0286$	$37.6940$	$149.4998$	$498.2002$	$1448.2716$
$N = 5$	$0.0000$	$0.8415$	$0.9093$	$0.1409$	$-0.7668$	$-1.1336$	$-2.0603$	$-11.8422$	$-65.6961$	$-287.9615$	$-1056.9392$
$N = 6$	$0.0000$	$0.8415$	$0.9093$	$0.1411$	$-0.7560$	$-0.9376$	$0.0372$	$3.7172$	$22.5894$	$120.2379$	$548.9652$
$N = 7$	$0.0000$	$0.8415$	$0.9093$	$0.1411$	$-0.7568$	$-0.9609$	$-0.3224$	$0.0867$	$-4.3167$	$-37.2105$	$-215.7512$
$N = 8$	$0.0000$	$0.8415$	$0.9093$	$0.1411$	$-0.7568$	$-0.9588$	$-0.2748$	$0.7407$	$2.0142$	$9.6767$	$65.3945$
$N = 9$	$0.0000$	$0.8415$	$0.9093$	$0.1411$	$-0.7568$	$-0.9589$	$-0.2798$	$0.6470$	$0.8294$	$-1.4281$	$-16.8119$
$N = 10$	$0.0000$	$0.8415$	$0.9093$	$0.1411$	$-0.7568$	$-0.9589$	$-0.2794$	$0.6580$	$1.0100$	$0.7135$	$2.7611$

If you examine the table of values, then you’ll notice that mystery(x,N) eventually stops changing as a function of $N$. That is, the value of mystery(x,N) is essentially the same for all large values of $N$. We call this the limiting value of the function. To be concrete, we say that two numbers $a$ and $b$ are essentially the same if they differ by less than $10^{-10}$. $$ |a - b| < 10^{-10} = 0. 000 000 000 1 $$ Create a new Python function limit(x) which calculates the limiting value of mystery(x,N) for each $x$. You will want to use the built-in Python function abs() to compute the absolute value.
Use Python to approximate the smallest and biggest values of mystery(x,N) on the domain $-3 \leq x \leq 3$ when $N$ is big. To do so, calculate the value of limit(x) at a large number of points in the domain $-3 \leq x \leq 3$. Your final answers for the biggest and smallest values should specify both: the value of limit(x) and the x where this value is obtained.
On paper: If you look carefully at the code, you’ll notice that mystery(x,N) is a sum of powers of $x$. Write it out as a polynomial, for $N = 3$. To do so, you are going to need to use the factorial function. The function math.factorial(n) computes the product of the first $n$ whole numbers. For example, $$ \texttt{math.factorial(4)} = 1 \cdot 2 \cdot 3 \cdot 4 = 24. $$ In mathematics, we usually write the factorial function as $n!$ because it grows so quickly. It’s a very exciting function!
In Desmos: Graph the function mystery(x,N) with a slider for $N$. Notice that mystery(x,N) is a summation from $n = 0$ to $n = N$. In Desmos, typing sum produces a summation sign $\displaystyle \sum_{n=}$.
What is the mystery function calculating?

Acknowledgements

Thanks to Mr. Nanthivarman for inviting me to speak at Lester B Pearson Collegiate Institute. Thanks to the following people for feedback on early drafts of this talk.

Infinite Limits in Python

What is this function calculating?

Acknowledgements

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