如何使用交替数学操作的python上的for循环？

Question

我需要编写一个代码，首先使用级数展开计算对数的近似，然后显示这个近似，以及近似结果和实际对数结果之间的差异。所有的数字都必须四舍五入到小数点6位。

作为输入，我收到的值是我必须找到对数的数字，以及我想要使用的对数级数展开的多少项。

首先，我必须定义并调用主函数，然后初始化任何变量，然后提示用户输入值，然后设置一个for -循环来执行级数展开，然后计算实际的对数答案，然后计算差额。最后，我必须打印近似值和差值。我还需要确保包括每一步的评论。

实数x (其中0

log(x) = (x-1) - (1/2)(x-1)^2 + (1/3)(x-1)^3 - (1/4)(x-1)^4 +.

我解决的问题是如何使用for循环来交替加法和减法。我也不确定我的其余代码是否编写正确，但它正在运行。不过，我从中得到的答案并不是正确的数学。

下面是我到目前为止编写的代码：

# This program displays the approximate logarithm and the difference between that and the actual logarithm of a number inputted by the user to six decimal places. 
# The definition of the main function
def main():
    import math
    # Initialize any necessary variables and prompt the user for values
    num = eval(input("Enter the number whose logarithm you want to be computed: "))
    term = eval(input("Enter how many terms of the series expansion for logarithms you want to use: "))
    approx = 0
    
    # Setup a loop based on the values entered by the user
    # Calculate an approximation to the logarithm of the number entered by the user
    for i in range(2, term+1):
        if i % 2 == 0:
            approx += (num-1) - ((1/i)*(num-1)**i)
        else:
            approx += (num-1) - ((1/i)*(num-1)**i) + ((1/(i+1))*((num-1)**(i+1))
    rapprox = round(approx, 6)
 
    # Compute logarithm and difference
    mog = math.log(num)
    rmog = round(mog, 6)
    diffy = rapprox - rmog

    # Display the computed values
    print("The computed approximation to log", num, "is: ", rapprox)
    print("The difference between the approximation and the actual value for log", num, "is: ", diffy)

    # Calling the main function 
    if __name__ == "__main__":
        main()

我们得到了两个样本输出，如下所示：

    Please enter the number whose logarithm you want to be computed: 1.5
    Please enter how many terms of the series expansion for logarithm you want to use: 5
    The computed approximation to log 1.5 is: 0.407292
    The difference between the approximation and the actual value for log 1.5 is: 0.001827

这是：

    Please enter the number whose logarithm you want to be computed: 0.8
    Please enter how many terms of the series expansion for logarithm you want to use: 4
    The computed approximation to log 0.8 is: -0.223067
    The difference between the approximation and the actual value for log 0.8 is: 0.000077

如果你需要更多的信息，因为这是不清楚的问题，我可以澄清，谢谢！

Answer 1

请注意，

log(x) = (x-1) - (1/2)(x-1)^2 + (1/3)(x-1)^3 - (1/4)(x-1)^4 +

= -(1-x) - (1/2)(1-x)^2 - (1/3)(1-x)^3 - (1/4)(1-x)^4 -

= - ( 1-x) + (1/2)(1-x)^2 + (1/3)(1-x)^3 + (1/4)(1-x)^4 +.

因此，可以将对数的近似实现为

def log_approx(x, N):
    """Computes an approximation of log(x) using N terms"""
    s = 0.0
    for n in range(1, N+1):
        s -= (1-x)**n / n
    return s

或者更短

def log_approx(x, N):
    """Computes an approximation of log(x) using N terms"""
    return -sum(((1-x)**n / n for n in range(1, N+1)))