## 2019年10月14日月曜日

### 数学 - Python - 解析学 - 級数 - べき級数 - 対数関数、指数関数、係数、累乗根、収束半径、絶対値、極限、逆数

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

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, log

print('27.')

n, m, x = symbols('n, m, x')
an = n / log(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('sample27.png')


% ./sample27.py
27.
__________
╱ │  n   │
lim n ╱  │──────│
n─→∞╲╱   │log(n)│

1

1

∞
____
╲
╲       n
╲   n⋅x
╱  ──────
╱   log(n)
╱
‾‾‾‾
n = 2

⎛    2      ⎞
⎜ 2⋅x       ⎟
⎜──────, red⎟
⎝log(2)     ⎠

⎛    3        2        ⎞
⎜ 3⋅x      2⋅x         ⎟
⎜────── + ──────, green⎟
⎝log(3)   log(2)       ⎠

⎛    4        3        2       ⎞
⎜ 4⋅x      3⋅x      2⋅x        ⎟
⎜────── + ────── + ──────, blue⎟
⎝log(4)   log(3)   log(2)      ⎠

⎛    5        4        3        2        ⎞
⎜ 5⋅x      4⋅x      3⋅x      2⋅x         ⎟
⎜────── + ────── + ────── + ──────, brown⎟
⎝log(5)   log(4)   log(3)   log(2)       ⎠

⎛    6        5        4        3        2         ⎞
⎜ 6⋅x      5⋅x      4⋅x      3⋅x      2⋅x          ⎟
⎜────── + ────── + ────── + ────── + ──────, orange⎟
⎝log(6)   log(5)   log(4)   log(3)   log(2)        ⎠

⎛    7        6        5        4        3        2         ⎞
⎜ 7⋅x      6⋅x      5⋅x      4⋅x      3⋅x      2⋅x          ⎟
⎜────── + ────── + ────── + ────── + ────── + ──────, purple⎟
⎝log(7)   log(6)   log(5)   log(4)   log(3)   log(2)        ⎠

⎛    8        7        6        5        4        3        2       ⎞
⎜ 8⋅x      7⋅x      6⋅x      5⋅x      4⋅x      3⋅x      2⋅x        ⎟
⎜────── + ────── + ────── + ────── + ────── + ────── + ──────, pink⎟
⎝log(8)   log(7)   log(6)   log(5)   log(4)   log(3)   log(2)      ⎠

⎛    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⎟
⎝log(9)   log(8)   log(7)   log(6)   log(5)   log(4)   log(3)   log(2)      ⎠

⎛     10       9        8        7        6        5        4        3
⎜ 10⋅x      9⋅x      8⋅x      7⋅x      6⋅x      5⋅x      4⋅x      3⋅x      2⋅x
⎜─────── + ────── + ────── + ────── + ────── + ────── + ────── + ────── + ────
⎝log(10)   log(9)   log(8)   log(7)   log(6)   log(5)   log(4)   log(3)   log(

2          ⎞
⎟
──, skyblue⎟
2)         ⎠

⎛     11        10       9        8        7        6        5        4
⎜ 11⋅x      10⋅x      9⋅x      8⋅x      7⋅x      6⋅x      5⋅x      4⋅x      3⋅
⎜─────── + ─────── + ────── + ────── + ────── + ────── + ────── + ────── + ───
⎝log(11)   log(10)   log(9)   log(8)   log(7)   log(6)   log(5)   log(4)   log

3        2         ⎞
x      2⋅x          ⎟
─── + ──────, yellow⎟
(3)   log(2)        ⎠

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