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//Title of this code
//Rextester.Program.Main is the entry point for your code. Don't change it.

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text.RegularExpressions;

namespace Rextester
{
    public class Program
    {
        //NOTE: If you try setting taylorX = maclaurinX you will get an exception error because of the different Radius of Convergence
        public static void Main(string[] args)
        {
            int n = 10; //How many terms we take the Taylor/Maclaurin series out to - increase for better precision
            double taylorX = -8.5; //Must be between -9 and -7 non-inclusive by Radius of Convergence
            double maclaurinX = 0.1; //Must be between -9 and 9 non-inclusive by Radius of Convergence
            
            TaylorSeries taylor = new TaylorSeries();
            taylor.TaylorApproximation(taylorX, n);
            
            MaclaurinSeries maclaurin = new MaclaurinSeries();
            maclaurin.MaclaurinApproximation(maclaurinX, n);
        }
    }
    
    public class TaylorSeries
    {
        public void TaylorApproximation(double x, int n)
        {
            double exact = 1/(9+x);
            double result = 0;
            if (x <= -9.0 || x >= -7)
                throw new System.ArgumentException("The Radius of Convergence for this Taylor series is (-9,-7) non-inclusive", "x");
            
            for (int i = 0; i < n; i++)
            {
                if (i % 2 == 0)
                {
                    result += Math.Pow(x + 8, i);
                }
                else
                {
                    result -= Math.Pow(x + 8, i);
                }
            }
            
            Console.WriteLine("The Taylor approximation is {0}", result);
            Console.WriteLine("The exact value is {0}", exact);
            Console.WriteLine("There is an error of {0}", Math.Abs(exact - result));
            Console.WriteLine();
        }
    }
    
    public class MaclaurinSeries
    {
        public void MaclaurinApproximation(double x, int n)
        {
            double exact = 1/(9+x);
            double result = 0;
            if (x <= -9.0 || x >= 9.0)
                throw new System.ArgumentException("The Radius of Convergence for this Maclaurin series is (-9,9) non-inclusive", "x");
            
            for (int i = 0; i < n; i++)
            {
                if (i % 2 == 0)
                {
                    result += Math.Pow(x/9.0, i);
                }
                else
                {
                    result -= Math.Pow(x/9.0, i);
                }
            }
            
            result *= (1.0/9.0);
            
            Console.WriteLine("The Maclaurin approximation is {0}", result);
            Console.WriteLine("The exact value is {0}", exact);
            Console.WriteLine("There is an error of {0}", Math.Abs(exact - result));
            Console.WriteLine();
        }
    }
}

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Compilation time: 0.12 sec, absolute running time: 0.21 sec, cpu time: 0.19 sec, average memory usage: 17 Mb, average nr of threads: 5 (cached)