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Writing C Code to Compute the Maclaurin Series Expansion for ( e^{-x} )

Writing C Code to Compute the Maclaurin Series Expansion for ( e^{-x} )

January 12, 2025 Education

How to Write C Code to Find the Sum of the First 10 Terms of ( e^{-x} ) Using the Maclaurin Series Expansion

The Maclaurin series expansion provides a powerful tool for approximating complex functions using a polynomial. For the function ( e^{-x} ), the Maclaurin series is given by:

( e^{-x} sum_{n0}^{infty} frac{-x^n}{n!} )

To compute the sum of the first 10 terms of this series, we use the formula and implement it in C programming. This article provides a comprehensive guide on writing C code to achieve this task.

C Programming for Maclaurin Series Expansion

The following C program implements the Maclaurin series expansion for ( e^{-x} ) and calculates the sum of the first 10 terms.

// Include necessary headers
#include stdio.h
#include math.h  // For pow and factorial functions
// Function to calculate factorial of a number
double factorial(int n) {
    double result  1.0;
    for (int i  1; i

Explanation of the Code

Factorial Function The `factorial` function calculates the factorial of a given integer `n`. This is used to compute the denominator of each term in the series expansion. Maclaurin Series Function The `exp_neg_x_maclaurin` function calculates the sum of the first 10 terms of the Maclaurin series for ( e^{-x} ). The summation starts from `k1` due to the 0-th term being 1. This reduces computational complexity. Main Function The `main` function prompts the user to enter a value for `x`, calculates the sum using the `exp_neg_x_maclaurin` function, and prints the result.

Usage: Compile the program using a C compiler, such as gcc, and run the executable. Input the desired value of `x`, and it will output the sum of the first 10 terms of the Maclaurin series for ( e^{-x} ).

Adaptive Approach for Series Calculation

Alternatively, you can use a more efficient approach to calculate the series terms, as described below. This method uses the previous term to find the current term, significantly reducing the computational complexity.

Let's denote the k-th term as ( T_k frac{-1^k x^k}{k!} ). The ratio between the k-th term and the (k-1)-th term is:

[ frac{T_k}{T_{k-1}} frac{-x}{k} ]

This allows you to compute each term using one additional multiplication and one division. Here's a MATLAB code example to demonstrate this:

function cumulative_sum  approximate_reciprocal_exponential(x)
    cumulative_sum  1;  // Initialize the sum with the 0-th term (1)
    series_term  1;  // Initialize the series term with the 0-th term (1)
    for k  1:9
        series_term  series_term * (-x / k);  // Compute the current term
        cumulative_sum  cumulative_sum   series_term;  // Add to the sum
    end
end

The C equivalent of this code would look similar. By following this strategy, you can efficiently compute the sum of the first 10 terms of the Maclaurin series for ( e^{-x} ).

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