## 2019年9月29日日曜日

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

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

よって収を半径は0。

コード

Python 3

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

print('10.')

n, m, x = symbols('n, m, x')
an = (n + 1) ** n
f = summation(an * x ** n, (n, 0, 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(10)
fs = [f.subs({m: m0}) for m0 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('sample10.png')


$./sample10.py 10. ____________ n ╱ │ n│ lim ╲╱ │(n + 1) │ n─→∞ ∞ 0 ∞ ___ ╲ ╲ n n ╱ x ⋅(n + 1) ╱ ‾‾‾ n = 0 ⎛ 0 ⎞ ⎜ ___ ⎟ ⎜ ╲ ⎟ ⎜ ╲ n n ⎟ ⎜ ╱ x ⋅(n + 1) , red⎟ ⎜ ╱ ⎟ ⎜ ‾‾‾ ⎟ ⎝n = 0 ⎠ ⎛ 1 ⎞ ⎜ ___ ⎟ ⎜ ╲ ⎟ ⎜ ╲ n n ⎟ ⎜ ╱ x ⋅(n + 1) , green⎟ ⎜ ╱ ⎟ ⎜ ‾‾‾ ⎟ ⎝n = 0 ⎠ ⎛ 2 ⎞ ⎜ ___ ⎟ ⎜ ╲ ⎟ ⎜ ╲ n n ⎟ ⎜ ╱ x ⋅(n + 1) , blue⎟ ⎜ ╱ ⎟ ⎜ ‾‾‾ ⎟ ⎝n = 0 ⎠ ⎛ 3 ⎞ ⎜ ___ ⎟ ⎜ ╲ ⎟ ⎜ ╲ n n ⎟ ⎜ ╱ x ⋅(n + 1) , brown⎟ ⎜ ╱ ⎟ ⎜ ‾‾‾ ⎟ ⎝n = 0 ⎠ ⎛ 4 ⎞ ⎜ ___ ⎟ ⎜ ╲ ⎟ ⎜ ╲ n n ⎟ ⎜ ╱ x ⋅(n + 1) , orange⎟ ⎜ ╱ ⎟ ⎜ ‾‾‾ ⎟ ⎝n = 0 ⎠ ⎛ 5 ⎞ ⎜ ___ ⎟ ⎜ ╲ ⎟ ⎜ ╲ n n ⎟ ⎜ ╱ x ⋅(n + 1) , purple⎟ ⎜ ╱ ⎟ ⎜ ‾‾‾ ⎟ ⎝n = 0 ⎠ ⎛ 6 ⎞ ⎜ ___ ⎟ ⎜ ╲ ⎟ ⎜ ╲ n n ⎟ ⎜ ╱ x ⋅(n + 1) , pink⎟ ⎜ ╱ ⎟ ⎜ ‾‾‾ ⎟ ⎝n = 0 ⎠ ⎛ 7 ⎞ ⎜ ___ ⎟ ⎜ ╲ ⎟ ⎜ ╲ n n ⎟ ⎜ ╱ x ⋅(n + 1) , gray⎟ ⎜ ╱ ⎟ ⎜ ‾‾‾ ⎟ ⎝n = 0 ⎠ ⎛ 8 ⎞ ⎜ ___ ⎟ ⎜ ╲ ⎟ ⎜ ╲ n n ⎟ ⎜ ╱ x ⋅(n + 1) , skyblue⎟ ⎜ ╱ ⎟ ⎜ ‾‾‾ ⎟ ⎝n = 0 ⎠ ⎛ 9 ⎞ ⎜ ___ ⎟ ⎜ ╲ ⎟ ⎜ ╲ n n ⎟ ⎜ ╱ x ⋅(n + 1) , yellow⎟ ⎜ ╱ ⎟ ⎜ ‾‾‾ ⎟ ⎝n = 0 ⎠$