Program Overview
This Python program reads two values: x and n.
It then calculates the sum of the first n terms of the following alternating series:
$$
S_n = \sum_{i=1}^{n} \frac{(-1)^{i+1} \cdot (2^i \cdot x^i)}{i + 2^i}
$$
Each term includes a power of x multiplied by 2^i, divided by the sum i + 2^i, and alternates in sign.
Python Code:
def custom_series(x, n):
total = 0
for i in range(1, n + 1):
numerator = (2 ** i) * (x ** i)
denominator = i + (2 ** i)
sign = (-1) ** (i + 1)
term = sign * numerator / denominator
total += term
return round(total, 6)
# Run the program
x = float(input("Enter x: "))
n = int(input("Enter number of terms n: "))
result = custom_series(x, n)
print(f"Result of the series: {result}")
Sample Output:
Enter x: 2
Enter number of terms n: 3
Result of the series: 2.133333
Step-by-Step Explanation:
- The numerator of each term is 2^i × x^i
- The denominator is i + 2^i
- The sign alternates using (-1)^{i+1}
- Each term is added to the total, and the final result is rounded to 6 decimal places