## 2019年10月10日木曜日

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

1. $\begin{array}{l}\underset{n\to \infty }{\mathrm{lim}}{\left|n\right|}^{\frac{1}{n}}\\ =1\end{array}$

よって、収束半径は1。

コード

Python 3

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

print('23.')

n, m, x = symbols('n, m, x')
an = n
f = summation(an * x ** n, (n, 2, m))

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

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

ms = range(3, 13)
# fs = [f.subs({m: m0}) for m0 in ms]

def g(m):
return sum([an.subs({n: m}) * x ** m for m in range(2, m)])

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

p = plot(*fs,
(x, -2, 2),
ylim=(-2, 2),
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('sample23.png')


$./sample23.py 23. n _____ lim ╲╱ │n│ n─→∞ 1 1 ⎧ ∞ for x = 1 ⎨ ⎩nan otherwise ⎛ 2 ⎞ ⎝2⋅x , red⎠ ⎛ 3 2 ⎞ ⎝3⋅x + 2⋅x , green⎠ ⎛ 4 3 2 ⎞ ⎝4⋅x + 3⋅x + 2⋅x , blue⎠ ⎛ 5 4 3 2 ⎞ ⎝5⋅x + 4⋅x + 3⋅x + 2⋅x , brown⎠ ⎛ 6 5 4 3 2 ⎞ ⎝6⋅x + 5⋅x + 4⋅x + 3⋅x + 2⋅x , orange⎠ ⎛ 7 6 5 4 3 2 ⎞ ⎝7⋅x + 6⋅x + 5⋅x + 4⋅x + 3⋅x + 2⋅x , purple⎠ ⎛ 8 7 6 5 4 3 2 ⎞ ⎝8⋅x + 7⋅x + 6⋅x + 5⋅x + 4⋅x + 3⋅x + 2⋅x , pink⎠ ⎛ 9 8 7 6 5 4 3 2 ⎞ ⎝9⋅x + 8⋅x + 7⋅x + 6⋅x + 5⋅x + 4⋅x + 3⋅x + 2⋅x , gray⎠ ⎛ 10 9 8 7 6 5 4 3 2 ⎞ ⎝10⋅x + 9⋅x + 8⋅x + 7⋅x + 6⋅x + 5⋅x + 4⋅x + 3⋅x + 2⋅x , skyblue⎠ ⎛ 11 10 9 8 7 6 5 4 3 2 ⎝11⋅x + 10⋅x + 9⋅x + 8⋅x + 7⋅x + 6⋅x + 5⋅x + 4⋅x + 3⋅x + 2⋅x , yell ⎞ ow⎠$