## 2019年10月12日土曜日

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

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

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('25.')

n, m, x = symbols('n, m, x')
an = n ** 2
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(2, 12)
# 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('sample25.png')


% ./sample25.py
25.
______
n ╱ │ 2│
lim ╲╱  │n │
n─→∞

1

1

⎧ ∞   for x = 1
⎨
⎩nan  otherwise

(x, red)

⎛   2           ⎞
⎝4⋅x  + x, green⎠

⎛   3      2          ⎞
⎝9⋅x  + 4⋅x  + x, blue⎠

⎛    4      3      2           ⎞
⎝16⋅x  + 9⋅x  + 4⋅x  + x, brown⎠

⎛    5       4      3      2            ⎞
⎝25⋅x  + 16⋅x  + 9⋅x  + 4⋅x  + x, orange⎠

⎛    6       5       4      3      2            ⎞
⎝36⋅x  + 25⋅x  + 16⋅x  + 9⋅x  + 4⋅x  + x, purple⎠

⎛    7       6       5       4      3      2          ⎞
⎝49⋅x  + 36⋅x  + 25⋅x  + 16⋅x  + 9⋅x  + 4⋅x  + x, pink⎠

⎛    8       7       6       5       4      3      2          ⎞
⎝64⋅x  + 49⋅x  + 36⋅x  + 25⋅x  + 16⋅x  + 9⋅x  + 4⋅x  + x, gray⎠

⎛    9       8       7       6       5       4      3      2             ⎞
⎝81⋅x  + 64⋅x  + 49⋅x  + 36⋅x  + 25⋅x  + 16⋅x  + 9⋅x  + 4⋅x  + x, skyblue⎠

⎛     10       9       8       7       6       5       4      3      2
⎝100⋅x   + 81⋅x  + 64⋅x  + 49⋅x  + 36⋅x  + 25⋅x  + 16⋅x  + 9⋅x  + 4⋅x  + x, ye

⎞
llow⎠

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