## 2019年10月1日火曜日

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

1. $\begin{array}{l}\underset{n\to \infty }{\mathrm{lim}}{\left|\frac{1}{\sqrt{n}}\right|}^{\frac{1}{n}}\\ =\underset{n\to \infty }{\mathrm{lim}}\frac{1}{{n}^{\frac{1}{2n}}}\\ =1\end{array}$

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, sqrt

print('12.')

n, m, x = symbols('n, m, x')
an = 1 / sqrt(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})]:
pprint(o)
print()

ms = range(1, 11)
# 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, -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('sample12.png')


$./sample12.py 12. ______ ╱ │1 │ lim n ╱ │──│ n─→∞╲╱ │√n│ 1 1 ∞ ____ ╲ ╲ n ╲ x ╱ ── ╱ √n ╱ ‾‾‾‾ n = 1 (0, red) (x, green) ⎛ 2 ⎞ ⎜√2⋅x ⎟ ⎜───── + x, blue⎟ ⎝ 2 ⎠ ⎛ 3 2 ⎞ ⎜√3⋅x √2⋅x ⎟ ⎜───── + ───── + x, brown⎟ ⎝ 3 2 ⎠ ⎛ 4 3 2 ⎞ ⎜x √3⋅x √2⋅x ⎟ ⎜── + ───── + ───── + x, orange⎟ ⎝2 3 2 ⎠ ⎛ 5 4 3 2 ⎞ ⎜√5⋅x x √3⋅x √2⋅x ⎟ ⎜───── + ── + ───── + ───── + x, purple⎟ ⎝ 5 2 3 2 ⎠ ⎛ 6 5 4 3 2 ⎞ ⎜√6⋅x √5⋅x x √3⋅x √2⋅x ⎟ ⎜───── + ───── + ── + ───── + ───── + x, pink⎟ ⎝ 6 5 2 3 2 ⎠ ⎛ 7 6 5 4 3 2 ⎞ ⎜√7⋅x √6⋅x √5⋅x x √3⋅x √2⋅x ⎟ ⎜───── + ───── + ───── + ── + ───── + ───── + x, gray⎟ ⎝ 7 6 5 2 3 2 ⎠ ⎛ 8 7 6 5 4 3 2 ⎞ ⎜√2⋅x √7⋅x √6⋅x √5⋅x x √3⋅x √2⋅x ⎟ ⎜───── + ───── + ───── + ───── + ── + ───── + ───── + x, skyblue⎟ ⎝ 4 7 6 5 2 3 2 ⎠ ⎛ 9 8 7 6 5 4 3 2 ⎞ ⎜x √2⋅x √7⋅x √6⋅x √5⋅x x √3⋅x √2⋅x ⎟ ⎜── + ───── + ───── + ───── + ───── + ── + ───── + ───── + x, yellow⎟ ⎝3 4 7 6 5 2 3 2 ⎠$