2019年10月17日木曜日

数学 - Python - 解析学 - 級数 - べき級数 - 階乗、係数、累乗根、収束半径、絶対値、極限、逆数

1. $\begin{array}{l}\underset{n\to \infty }{\mathrm{lim}}{\left|\frac{{\left(-1\right)}^{n}+1}{n!}\right|}^{\frac{1}{n}}\\ =0\end{array}$

よって、収束半径は

$\infty$

コード

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, summation, oo, Limit, plot, factorial

print('30.')

n, m, x = symbols('n, m, x')
n, m = symbols('n, m', integer=True, positive=True)

an = ((-1) ** n + 1) / factorial(n)
f = summation(an * x ** n, (n, 1, m))

s = Limit(abs(an) ** (1 / n), n, oo)

# for o in [s,  s.doit(), 1 / s.doit(), f.subs({m: oo})]:
for o in [s, f.subs({m: oo})]:
pprint(o)
print()

ms = range(1, 11, 2)
# fs = [f.subs({m: m0}) for m0 in ms]

def g(m):
return sum([an.subs({n: m}) * x ** m for m in range(1, m)])

fs = [g(m) for m in ms]

p = plot(*fs,
(x, -10, 10),
ylim=(0, 20),
legend=False,
show=False)
colors = ['red', 'green', 'blue', 'brown', 'orange',
'purple', 'pink', 'gray', 'skyblue', 'yellow']

for s, color in zip(p, colors):
s.line_color = color

for o in zip(fs, colors):
pprint(o)
print()

p.show()
p.save('sample30.png')


% ./sample30.py
30.
_____________
╱ │    n    │
╱  │(-1)  + 1│
lim n ╱   ───────────
n─→∞╲╱         n!

nan

(0, red)

⎛ 2       ⎞
⎝x , green⎠

⎛ 4           ⎞
⎜x     2      ⎟
⎜── + x , blue⎟
⎝12           ⎠

⎛  6    4            ⎞
⎜ x    x     2       ⎟
⎜─── + ── + x , brown⎟
⎝360   12            ⎠

⎛   8      6    4             ⎞
⎜  x      x    x     2        ⎟
⎜───── + ─── + ── + x , orange⎟
⎝20160   360   12             ⎠

%