Power series expansion of e, gifified.

README.md

Q

One way to express $e$ is via a power series:

$$e = \sum_{n=0}^{\infty} \frac{x^n}{n!}$$

If we plot the first k terms of this series for increasing x we observe a normal distribution of the terms.

expansion.py

#!/usr/bin/env python
# coding: utf-8

from __future__ import division
from pandas import Series
from math import factorial
import matplotlib.pyplot as plt
import seaborn as sns

def a_x(x, n):
    return x**n / factorial(n)

def plot_exp_x_up_to_n(x, n=10):
    s = Series((a_x(x, i) for i in range(n)))
    with sns.color_palette("Greys_r"):
        s.plot(kind='bar')

        plt.title('terms of e(x) series for x = %d' % x)
        plt.minorticks_off()

        plt.xlabel('term n')
        plt.savefig('x_%04d_n_%04d.png' % (x, n))

if __name__ == '__main__':
    for i in range(0, 40):
        plot_exp_x_up_to_n(i, 40)

Makefile

SHELL = /bin/bash

images:
	python expansion.py

gifs:
	for fn in *png; do convert "$$fn" "$${fn%.png}.gif"; done

anim.gif:
	gifsicle --colors 16 --delay=35 --loop x_*.gif > anim.gif 

clean:
	rm -f x_*_n_*.png
	rm -f x_*_n_*.gif
	rm -f anim.gif