Blog Entry (c) Saturday, October 31, 2026, by James Pate Williams, Jr. Simple Maclaurin and Taylor Series Expansions

// MaclaurinSeries.cpp : This file contains the 'main' function.
// Program execution begins and ends there.
// Copyright (c) Saturday, October 31, 2026
// by James Pate Williams, Jr., BA, BS, MSwE, PhD

#include <cmath>
#include <iomanip>
#include <iostream>

// iterative n-factorial function
static double iFactorial(int n)
{
    double factorial = 1.0;

    for (int i = 2; i <= n; i++)
        factorial *= i;

    return factorial;
}

static double Series1(double x, int n)
{
    double x2 = x * x, x4 = x2 * x2, x6 = x2 * x4;

    switch (n)
    {
    case 0:
        return 1.0;
    case 1:
        return 1.0 - x2;
    case 2:
        return 1.0 - x2 + 9 * x4 / iFactorial(4);
    case 3:
        return 1.0 - x2 + 9 * x4 / iFactorial(4) - 225 * x6 / iFactorial(6);
    case 4:
        return 1.0 - x2 + 9 * x4 / iFactorial(4) - 225 * x6 / iFactorial(6) +
            10825 * x6 * x2 / iFactorial(8);
    default:
        return 0.0;
    }
}

static double Series2(double x, int n)
{
    double y = x - 1;
    double y2 = y * y, y3 = y * y2, y4 = y * y3, y5 = y * y4;
    double y6 = y * y5, y7 = y * y6, y8 = y * y7, y9 = y * y8;
    double y10 = y * y9;

    switch (n)
    {
    case 0:
        return log(1.0);
    case 1:
        return log(1.0) + y;
    case 2:
        return log(1.0) + y - y2 / iFactorial(2);
    case 3:
        return log(1.0) + y - y2 / iFactorial(2) + 2.0 * y3 / iFactorial(3);
    case 4:
        return log(1.0) + y - y2 / iFactorial(2) + 2.0 * y3 / iFactorial(3)
            - 6.0 * y4 / iFactorial(4);
    case 5:
        return log(1.0) + y - y2 / iFactorial(2) + 2.0 * y3 / iFactorial(3)
            - 6.0 * y4 / iFactorial(4) + 24.0 * y5 / iFactorial(5);
    case 6:
        return log(1.0) + y - y2 / iFactorial(2) + 2.0 * y3 / iFactorial(3)
            - 6.0 * y4 / iFactorial(4) + 24.0 * y5 / iFactorial(5)
            - 120.0 * y6 / iFactorial(6);
    case 7:
        return log(1.0) + y - y2 / iFactorial(2) + 2.0 * y3 / iFactorial(3)
            - 6.0 * y4 / iFactorial(4) + 24.0 * y5 / iFactorial(5)
            - 120.0 * y6 / iFactorial(6) + 720.0 * y7 / iFactorial(7);
    case 8:
        return log(1.0) + y - y2 / iFactorial(2) + 2.0 * y3 / iFactorial(3)
            - 6.0 * y4 / iFactorial(4) + 24.0 * y5 / iFactorial(5)
            - 120.0 * y6 / iFactorial(6) + 720.0 * y7 / iFactorial(7)
            - 5040.0 * y8 / iFactorial(8);
    case 9:
        return log(1.0) + y - y2 / iFactorial(2) + 2.0 * y3 / iFactorial(3)
            - 6.0 * y4 / iFactorial(4) + 24.0 * y5 / iFactorial(5)
            - 120.0 * y6 / iFactorial(6) + 720.0 * y7 / iFactorial(7)
            - 5040.0 * y8 / iFactorial(8) + 40320 * y9 / iFactorial(9);
    case 10:
        return log(1.0) + y - y2 / iFactorial(2) + 2.0 * y3 / iFactorial(3)
            - 6.0 * y4 / iFactorial(4) + 24.0 * y5 / iFactorial(5)
            - 120.0 * y6 / iFactorial(6) + 720.0 * y7 / iFactorial(7)
            - 5040.0 * y8 / iFactorial(8) + 40320 * y9 / iFactorial(9)
            - 362880 * y10 / iFactorial(10);
    default:
    {
        double sign = 1.0, sum = log(1.0);

        for (int i = 1; i <= n; i++)
        {
            sum += sign * pow(y, i) / i;
            sign *= -1.0;
        }

        return sum;
    }
    }
}

int main()
{
    std::cout << "n" << '\t' << "n!" << std::endl;

    for (int n = 0; n <= 10; n++)
        std::cout << n << '\t' << iFactorial(n) << std::endl;

    double x = 0.015;
    std::cout << std::fixed << std::setprecision(10);
    std::cout << "cos(arctan(x)) = (1 + x^2)^-1/2" << std::endl;
    std::cout << "Maclaurin Series Expansion of (1 + x^2)^-1/2" << std::endl;
    std::cout << "# is the number + 1 of nonzero terms in the summation\r\n";
    std::cout << "x = " << x << std::endl;
    std::cout << "#" << '\t' << "Series" << "\t\t" << "(1 + x^2)^-1/2";
    std::cout << std::endl;

    for (int n = 0; n <= 4; n++)
    {
        std::cout << n << '\t';
        std::cout << std::fixed << std::setprecision(10);
        std::cout << Series1(x, n) << '\t';
        std::cout << pow(1.0 + x * x, -0.5) << std::endl;
    }

    x = 0.125;
    std::cout << "Maclaurin Series Expansion of (1 + x^2)^-1/2" << std::endl;
    std::cout << "x = " << x << std::endl;
    std::cout << "#" << '\t' << "Series" << "\t\t" << "(1 + x^2)^-1/2";
    std::cout << std::endl;

    for (int n = 0; n <= 4; n++)
    {
        std::cout << n << '\t';
        std::cout << std::fixed << std::setprecision(10);
        std::cout << Series1(x, n) << '\t';
        std::cout << pow(1.0 + x * x, -0.5) << std::endl;
    }

    x = 0.5;
    std::cout << "Maclaurin Series Expansion of (1 + x^2)^-1/2" << std::endl;
    std::cout << "x = " << x << std::endl;
    std::cout << "#" << '\t' << "Series" << "\t\t" << "(1 + x^2)^-1/2";
    std::cout << std::endl;

    for (int n = 0; n <= 4; n++)
    {
        std::cout << n << '\t';
        std::cout << std::fixed << std::setprecision(10);
        std::cout << Series1(x, n) << '\t';
        std::cout << pow(1.0 + x * x, -0.5) << std::endl;
    }

    x = 1.25;
    std::cout << "Taylor Series Expansion of log(x)" << std::endl;
    std::cout << "About x = 1" << std::endl;
    std::cout << "x = " << x << std::endl;
    std::cout << "#" << '\t' << "Series" << "\t\t" << "ln(x)";
    std::cout << std::endl;

    for (int n = 0; n <= 15; n++)
    {
        std::cout << n << '\t';
        std::cout << std::fixed << std::setprecision(10);
        std::cout << Series2(x, n) << '\t';
        std::cout << std::log(x) << std::endl;
    }

    x = 1.5;
    std::cout << "Taylor Series Expansion of log(x)" << std::endl;
    std::cout << "About x = 1" << std::endl;
    std::cout << "x = " << x << std::endl;
    std::cout << "#" << '\t' << "Series" << "\t\t" << "ln(x)";
    std::cout << std::endl;

    for (int n = 0; n <= 30; n++)
    {
        std::cout << n << '\t';
        std::cout << std::fixed << std::setprecision(10);
        std::cout << Series2(x, n) << '\t';
        std::cout << std::log(x) << std::endl;
    }

    return 0;
}