Step by Step solution about How to resolve the algorithm Euler's constant 0.5772... step by step in the Python programming language

The provided Python code calculates Euler's constant (also known as the Euler-Mascheroni constant) using Euler's Zeta Series. Here's how it works:

1. Importing Libraries:

from scipy import special as s

This line imports the special submodule from the scipy library, which provides functions for mathematical operations, including the Zeta function used in this code.

2. Function Definition:

def eulers_constant(n):

The code defines a function called eulers_constant that takes a positive integer n as its argument. This function will calculate and return an approximation of Euler's constant.

3. Initialization:

   k = 2
   euler = 0

Inside the function, two variables are initialized:

• k is set to 2, which represents the starting value of the loop to calculate the series.
• euler is set to 0, which will accumulate the sum of the series terms.

4. Loop to Calculate the Series:

   while k <= n:
       euler += (s.zeta(k) - 1)/k
       k += 1

This loop iterates from k=2 to k=n. In each iteration, it calculates the k-th term of the series:

• s.zeta(k) computes the Zeta function for the given k.
• (s.zeta(k) - 1)/k subtracts 1 from the Zeta function and divides it by k.
• This computed value is added to the accumulating euler variable.

5. Final Calculation:

   return 1 - euler

After the loop finishes, the function calculates Euler's constant by subtracting the accumulated euler value from 1. The result is then returned.

6. Example:

print(eulers_constant(47))

This line demonstrates the usage of the function by printing the approximation of Euler's constant for n=47. When executed, it will output the calculated value, which should be close to the actual value of Euler's constant, approximately 0.5772.

In summary, this code uses a loop to calculate a series of terms based on the Zeta function and accumulates them to approximate Euler's constant. By providing a value for n, the code can calculate the approximation of Euler's constant to a desired level of accuracy.

# /**************************************************
# Subject: Computation of Euler's constant 0.5772...
#          with Euler's Zeta Series.
# tested : Python 3.11 
# -------------------------------------------------*/

from scipy import special as s

def eulers_constant(n):
    k = 2
    euler = 0
    while k <= n:
        euler += (s.zeta(k) - 1)/k
        k += 1
    return 1 - euler

print(eulers_constant(47))