python - 使用 Sympy，计算其项交替符号的序列的限制

Question

我正在尝试通过 Sympy 找到这个序列的极限

               ((-1)**(n))*(2**(-n))

使用 PyCharm 调试器和 IPython 控制台。

python 模块是:

import numpy as np
from plotly.graph_objs import Scatter, Figure
from sympy import Symbol, srepr, limit_seq, oo
from sympy.abc import n

import plotly.plotly as py

py.sign_in('****', '****************')



sym_func = 1 + ((-1)**(n))*(2**(-n))

def rng_func(i):

    try:
        return 1 + np.power(-1, i)/(np.power(2,i))
    except ZeroDivisionError as e:
        raise ValueError('Invalid inputs') from e


class Sequence(object):

    def __init__(self, i):

        self.sum = 0
        self.dom = np.arange(1, i + 1, 1)
        self.rng = np.array([])
        self.is_positive = True
        self.is_alternating = True

        for i in self.dom:
            rng_val = rng_func(i)
            self.rng = np.append(self.rng, [rng_val])  
            self.sum += rng_val

        sign_array = np.sign(self.rng)

    for i in self.dom:

        if self.rng[i - 1] <= 0:
            self.is_positive = False
            break

    for i in self.dom:

        if sign_array[0] == -1:
            alt_sign = np.power(-1, i)
        else:
            alt_sign = np.power(-1, i-1)

        if alt_sign != sign_array[i-1]:
            self.is_alternating = False
            break


seq = Sequence(10)

data = [Scatter(x=seq.dom, y=seq.rng)]
fig = Figure(data=data)

py.image.save_as(fig, filename='plots/I3.png')  

IPython 控制台:

Connected to pydev debugger (build 171.4424.56)
Backend Qt5Agg is interactive backend. Turning interactive mode on.
import sys; print('Python %s on %s' % (sys.version, sys.platform))
Python 3.5.2 |Anaconda custom (64-bit)| (default, Jul  2 2016,    17:53:06) 
Type "copyright", "credits" or "license" for more information.
IPython 5.1.0 -- An enhanced Interactive Python.

In[1]: limit_seq(((-1)**(n))*(2**(-n)), n)
In[2]: limit_seq((1**(n))*(2**(-n)), n)
Out[3]: 
0

我的问题是，为什么 Sympy 不为 In[1]: 的限制返回一个值，而为 In[2]: 返回一个值？不同之处在于分子中的 1 与 -1。在 Sympy 中是否有不同的方法来创建交替符号？

Answer 1

截至目前(SymPy 1.1.1)，limit_seq 不支持带有 (-1)**n 的振荡序列。它的documentation以一种有点技术性的方式说这个:

The terms should be built from rational functions, indefinite sums, and indefinite products over an indeterminate n. A term is admissible if the scope of all product quantifiers are asymptotically positive. Every admissible term is asymptotically monotonous.

作为解决方法，您可以使用此方法:

def mylimit_seq(seq, n):
    n_ = Dummy("n", integer=True, positive=True)
    L1 = limit_seq(seq.subs(n, 2*n_), n_)
    L2 = limit_seq(seq.subs(n, 2*n_ + 1), n_)
    if L1 == L2:
        return L1

然后 mylimit_seq((-1)**n * 2**(-n), n) 返回 0。

---

在未来的版本中，limit_seq 将适用于此类序列。您现在还可以通过从 GitHub 克隆 SymPy 来获得此功能。 .