The factorial of a nonnegative integer n is written n ! (pronounced "n factorial") and is defined as follows: n ! = n · (n - 1) · (n - 2) · … · 1 (for values of n greater than or equal to 1) and n ! = 1 (for n = 0). For example, 5! = 5 · 4 · 3 · 2 · 1, which is 120. a) Write a program that reads a nonnegative integer and computes and prints its factorial.
b) Write a program that estimates the value of the mathematical constant e by using the formula:
e = 1 + 1/1! +1/2! + 1/3! +
c) Write a program that computes the value of by using the formula
e^x = 1 + x/1! + x^2/2! + x^3/3! +

Here are the programs. I am writing C++ and Python programs:

a)

C++

#include<iostream>   

using namespace std;    

int factorial(int number)  {  

   if (number == 0)  

       return 1;  

   return number * factorial(number - 1);  }    

int main()  {  

   int integer;

   cout<<"Enter a non negative integer: ";

   cin>>integer;

   cout<< "Factorial of "<< integer<<" is "<< factorial(integer)<< endl;  }

Python:

def factorial(number):  

   if number == 0:  

       return 1

   return number * factorial(number-1)  

integer = int(input("Enter a non negative integer: "))  

print("Factorial of", integer, "is", factorial(integer))

b)

C++

#include <iostream>  

using namespace std;

double factorial(int number) {  

if (number == 0)  

 return 1;  

return number * factorial(number - 1); }  

double estimate_e(int num){

    double e = 1;

    for(int i = 1; i < num; i++)

     e = e + 1/factorial(i);

     cout<<"e: "<< e; }  

int main(){

int term;

cout<<"Enter a term to evaluate: ";

cin>>term;

estimate_e(term);}

Python:

def factorial(number):  

   if number == 0:  

       return 1

   return number * factorial(number-1)  

def estimate_e(term):

   if not term:

       return 0

   else:

       return (1 / factorial(term-1)) + estimate_e(term-1)

number = int(input("Enter how many terms to evaluate "))

print("e: ", estimate_e(number))

c)

C++

#include <iostream>

using namespace std;

int main(){

   float terms, sumSeries, series;

   int i, number;

   cout << " Input the value of x: ";

   cin >> number;

   cout << " Input number of terms: ";

   cin >> terms;

   sumSeries = 1;

   series = 1;

   for (i = 1; i < terms; i++)      {

       series = series * number / (float)i;

       sumSeries = sumSeries + series;     }

   cout << " The sum  is : " << sumSeries << endl;  }  

Python    

def ePowerx(number,terms):

   sumSeries = 1

   series =1

   for x in range(1,terms):

       series = series * number / x;

       sumSeries = sumSeries + series;

   return sumSeries    

num = int(input("Enter a number: "))

term=int(input("Enter a number: "))

print("e^x: ",ePowerx(num,term))

Explanation:

a)

The program has a method factorial that takes numbers as parameter and computes the factorial of that number by using recursion. For example if the number = 3

The base case is  if (number == 0)

and

recursive is return number * factorial(number - 1);    

Since number = 3 is not equal to zero so the method calls itself recursively in order to return the factorial of 3

return 3* factorial(3- 1);

3 * factorial(2)

3* [2* factorial(2- 1) ]

3 * 2* [ factorial(1)]

3 * 2 * [1* factorial(1- 1) ]

3 * 2 * 1* [factorial(0)]

Now at factorial(0) the base condition is reached as number==0 So factorial(0) returns 1

Now the output is:

3 * 2 * 1* 1

return 6

So the output of this program is

Factorial of 3 is 6

b)

The method estimate_e takes a number i.e. num as parameter which represents the term and estimates the value of the mathematical constant e

The for loop iterates through each term. For example num = 3

then  

e = e + 1/factorial(i);  

The above statement calls works as:

e = 1 + 1/1! +1/2!

since the number of terms is 3

e is initialized to 1

i  is initialized to 1

So the statement becomes:

e = 1 + 1/factorial(1)

factorial function is called which returns 1 since factorial of 1 is 1 So,

e = 1 + 1/1

e = 2

Now at next iteration at i = 2 and e = 2

e = 2 + 1/factorial(2)

e = 2 + 1/2

e = 2.5

Now at next iteration at i = 3 and e =3

e = 3 + 1/factorial(3)

e = 3 + 1/6

e = 3.16666

So the output is:

e: 3.16666

c)

The program computes the sum of series using formula:

e^x = 1 + x/1! + x^2/2! + x^3/3! +...

The for loop iterates till the number of terms. Lets say in the above formula x is 2 and number of terms is 3. So the series become:

e^x = 1 + x/1! + x^2/2!

So number = 2

terms = 3

series =1

sumSeries = 1

i = 1

The statement series = series * number / (float)i; works as following:

series = 1 * 2 /1

series = 2

sumSeries = sumSeries + series;

sumSeries = 1 + 2

sumSeries = 3

At next iteration: i=2, series =2 , sumSeries =3

series = 2 * 2/2

series = 2

sumSeries = 3 + 2

sumSeries = 5

Now the loop breaks as i=3

So output returns the value of sumSeries i.e. 5

Output:

e^x: 5

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