## 2019年10月13日日曜日

### 数学 - Python - 解析学 - 級数 - べき級数 - 三角関数(余弦)、累乗、指数関数、係数、累乗根、収束半径、絶対値、極限、逆数

1. $\begin{array}{l}\underset{n\to \infty }{\mathrm{lim}}{\left|\frac{\mathrm{cos}{n}^{2}}{{n}^{n}}\right|}^{\frac{1}{n}}\\ =\underset{n\to \infty }{\mathrm{lim}}\frac{1}{n}\\ =0\end{array}$

よって収束半径は、

$\infty$

コード

Python 3

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

print('26.')

n, m, x = symbols('n, m, x')
an = cos(n ** 2) / n ** 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(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, -5, 5),
ylim=(-5, 5),
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('sample26.png')


% ./sample26.py
26.
_______________
n ╱ │ -n    ⎛ 2⎞│
lim ╲╱  │n  ⋅cos⎝n ⎠│
n─→∞

1

1

∞
___
╲
╲    -n  n    ⎛ 2⎞
╱   n  ⋅x ⋅cos⎝n ⎠
╱
‾‾‾
n = 1

(x⋅cos(1), red)

⎛ 2                         ⎞
⎜x ⋅cos(4)                  ⎟
⎜───────── + x⋅cos(1), green⎟
⎝    4                      ⎠

⎛ 3           2                        ⎞
⎜x ⋅cos(9)   x ⋅cos(4)                 ⎟
⎜───────── + ───────── + x⋅cos(1), blue⎟
⎝    27          4                     ⎠

⎛ 4            3           2                         ⎞
⎜x ⋅cos(16)   x ⋅cos(9)   x ⋅cos(4)                  ⎟
⎜────────── + ───────── + ───────── + x⋅cos(1), brown⎟
⎝   256           27          4                      ⎠

⎛ 5            4            3           2                          ⎞
⎜x ⋅cos(25)   x ⋅cos(16)   x ⋅cos(9)   x ⋅cos(4)                   ⎟
⎜────────── + ────────── + ───────── + ───────── + x⋅cos(1), orange⎟
⎝   3125         256           27          4                       ⎠

⎛ 6            5            4            3           2
⎜x ⋅cos(36)   x ⋅cos(25)   x ⋅cos(16)   x ⋅cos(9)   x ⋅cos(4)
⎜────────── + ────────── + ────────── + ───────── + ───────── + x⋅cos(1), purp
⎝  46656         3125         256           27          4

⎞
⎟
le⎟
⎠

⎛ 7            6            5            4            3           2
⎜x ⋅cos(49)   x ⋅cos(36)   x ⋅cos(25)   x ⋅cos(16)   x ⋅cos(9)   x ⋅cos(4)
⎜────────── + ────────── + ────────── + ────────── + ───────── + ───────── + x
⎝  823543       46656         3125         256           27          4

⎞
⎟
⋅cos(1), pink⎟
⎠

⎛ 8            7            6            5            4            3
⎜x ⋅cos(64)   x ⋅cos(49)   x ⋅cos(36)   x ⋅cos(25)   x ⋅cos(16)   x ⋅cos(9)
⎜────────── + ────────── + ────────── + ────────── + ────────── + ───────── +
⎝ 16777216      823543       46656         3125         256           27

2                        ⎞
x ⋅cos(4)                 ⎟
───────── + x⋅cos(1), gray⎟
4                     ⎠

⎛ 9            8            7            6            5            4
⎜x ⋅cos(81)   x ⋅cos(64)   x ⋅cos(49)   x ⋅cos(36)   x ⋅cos(25)   x ⋅cos(16)
⎜────────── + ────────── + ────────── + ────────── + ────────── + ────────── +
⎝387420489     16777216      823543       46656         3125         256

3           2                           ⎞
x ⋅cos(9)   x ⋅cos(4)                    ⎟
───────── + ───────── + x⋅cos(1), skyblue⎟
27          4                        ⎠

⎛ 10             9            8            7            6            5
⎜x  ⋅cos(100)   x ⋅cos(81)   x ⋅cos(64)   x ⋅cos(49)   x ⋅cos(36)   x ⋅cos(25)
⎜──────────── + ────────── + ────────── + ────────── + ────────── + ──────────
⎝10000000000    387420489     16777216      823543       46656         3125

4            3           2                          ⎞
x ⋅cos(16)   x ⋅cos(9)   x ⋅cos(4)                   ⎟
+ ────────── + ───────── + ───────── + x⋅cos(1), yellow⎟
256           27          4                       ⎠

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