数学 - Python - 解析学 - 数 - 数列と級数 - 数列の収束条件(2つの数列、和の上極限と上極限の和、不等式、等式、有限、負の無限大)

学習環境

解析入門(上) (松坂和夫数学入門シリーズ 4) (松坂和夫(著)、岩波書店)の第2章(数列と級数)、2.2(数列の収束条件)、問題8の解答を求めてみる。

- 上極限がすべて有限で等号が成り立たない例。
  
  $\begin{array}{l} a_{n} = {(- 1)}^{n} \\ b_{n} = {(- 1)}^{n + 1} \\ a_{n} + b_{n} = 0 \\ \underset{n \to a}{limsup} (a_{n} + b_{n}) = 0 \\ \underset{n \to \infty}{limsup} a_{n} + \underset{n \to \infty}{limsup} b_{n} \\ = 1 + 1 \\ = 2 \end{array}$
- 右辺の2つの上極限がともに有限で、左辺が負の無限大である2つの数列の例。
  
  $\begin{array}{l} a_{2 n - 1} = {(- 1)}^{2 n - 1} (2 n - 1) \\ a_{2 n} = 0 \\ b_{2 n - 1} = 0 \\ b_{2 n} = {(- 1)}^{2 n - 1} 2 n \\ \underset{n \to \infty}{limsup} a_{n} = 0 \\ \underset{n \to \infty}{limsup} b_{n} = 0 \\ a_{2 n - 1} + b_{2 n - 1} = {(- 1)}^{2 n - 1} (2 n - 1) = - (2 n - 1) \\ a_{2 n} + b_{2 a} = {(- 1)}^{2 n - 1} 2 n = - 2 n \\ \underset{n \to \infty}{limsup} (a_{n} + b_{n}) = - \infty \end{array}$

コード

Python 3

#!/usr/bin/env python3
import matplotlib.pyplot as plt

print('8.')


def a1(n):
    return (-1) ** n


def b1(n):
    return (-1) ** (n + 1)


def ab1(n):
    return a1(n) + b1(n)


def a2(n):
    if n % 2 == 1:
        return (-1) ** (2 * n - 1) * (2 * n - 1)
    return 0


def b2(n):
    if n % 2 == 1:
        return 0
    return (-1) ** (2 ** n - 1) * 2 * n


def ab2(n):
    return a2(n) + b2(n)


n = 10

for (a, b, ab) in [(a1, b1, ab1), (a2, b2, ab2)]:
    for n in range(1, n + 1):
        print(f'n = {n}, a{n} = {a(n)}, b{n} = {b(n)}, a{n} + b{n} = {ab(n)}')
    print()

plt.plot(range(1, n + 1), [a1(i) for i in range(1, n + 1)],
         range(1, n + 1), [b1(i) for i in range(1, n + 1)],
         range(1, n + 1), [ab1(i) for i in range(1, n + 1)],
         range(1, n + 1), [a2(i) for i in range(1, n + 1)],
         range(1, n + 1), [b2(i) for i in range(1, n + 1)],
         range(1, n + 1), [ab2(i) for i in range(1, n + 1)],
         marker='o')

plt.legend(['a1', 'b1', 'a1 + b1', 'a2', 'b2', 'a2 + b2'])
# plt.show()
plt.savefig('sample8.png')

入出力結果(cmd(コマンドプロンプト)、Terminal、Jupyter(IPython))

C:\Users\...>py sample8.py
8.
n = 1, a1 = -1, b1 = 1, a1 + b1 = 0
n = 2, a2 = 1, b2 = -1, a2 + b2 = 0
n = 3, a3 = -1, b3 = 1, a3 + b3 = 0
n = 4, a4 = 1, b4 = -1, a4 + b4 = 0
n = 5, a5 = -1, b5 = 1, a5 + b5 = 0
n = 6, a6 = 1, b6 = -1, a6 + b6 = 0
n = 7, a7 = -1, b7 = 1, a7 + b7 = 0
n = 8, a8 = 1, b8 = -1, a8 + b8 = 0
n = 9, a9 = -1, b9 = 1, a9 + b9 = 0
n = 10, a10 = 1, b10 = -1, a10 + b10 = 0

n = 1, a1 = -1, b1 = 0, a1 + b1 = -1
n = 2, a2 = 0, b2 = -4, a2 + b2 = -4
n = 3, a3 = -5, b3 = 0, a3 + b3 = -5
n = 4, a4 = 0, b4 = -8, a4 + b4 = -8
n = 5, a5 = -9, b5 = 0, a5 + b5 = -9
n = 6, a6 = 0, b6 = -12, a6 + b6 = -12
n = 7, a7 = -13, b7 = 0, a7 + b7 = -13
n = 8, a8 = 0, b8 = -16, a8 + b8 = -16
n = 9, a9 = -17, b9 = 0, a9 + b9 = -17
n = 10, a10 = 0, b10 = -20, a10 + b10 = -20


C:\Users\...>

Kamimura's blog

2019年4月30日火曜日

数学 - Python - 解析学 - 数 - 数列と級数 - 数列の収束条件(2つの数列、和の上極限と上極限の和、不等式、等式、有限、負の無限大)

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