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Program Overview

This Python program reads two values from the user:
- x: the base input value - n: the number of terms to compute
It then calculates the sum of the first n terms of the following custom series:
$$\sum_{k=1}^{n} \frac{x^k}{f(k)} \cdot s(k)$$ Where:
- f(k): the factorial used in the denominator (e.g. 2!, 4!, 8!, etc.) - s(k): the sign of the term (positive or negative) - The pattern of powers, factorials, and signs is manually defined in the code

Python Code:

import math

def custom_series(x, n):
    total = 0
    factorials = [1, 2, 4, 2, 8]  # Extendable pattern
    signs = [1, 1, 1, -1, 1]      # Alternating signs

    for k in range(n):
        power = k + 1
        fact = factorials[k] if k < len(factorials) else 1
        sign = signs[k] if k < len(signs) else 1
        term = sign * (x ** power) / math.factorial(fact)
        total += term

    return round(total, 6)

# Run the program
x = float(input("Enter value for x: "))
n = int(input("Enter number of terms: "))
result = custom_series(x, n)
print(f"Series sum: {result}")

Sample Output:

Enter value for x: 2  
Enter number of terms: 5  

Series sum: 4.933333

Step-by-Step Explanation:

- The user inputs x and n
- Each term is calculated using a specific power, factorial, and sign
- The term is added to the running total
- The final result is rounded to 6 decimal places and displayed