Reference no: EM132360780

Question

PYTHON

Write a program that computes the value of e using all three of the methods described. Use values of 1, 2, 5, 10, 100, 1000, and 10000 for the first method.

Values of n = 1,2,3,4,5,6,7,8,9,10, and 20 for the second method.

Values of n = 1, 3, 5, 7, and 9 for the third method.

Your program must print out the value of e computed with each value of n for each method and the difference with the value of in math library. Make certain that the outputs are labeled as to which method is being used and what value of n was used.

METHOD #1

For example, the value of (1 + 1/n)n approaches e as n gets bigger and bigger:

graph of (1+1/n)^n

n

(1 + 1/n)n

1

2.00000

2

2.25000

5

2.48832

10

2.59374

100

2.70481

1,000

2.71692

10,000

2.71815

100,000

2.71827

In the equations that follow, "!" means factorial.

0! = 1, 1! =1, 2! = 2·1 = 2, 3! = 3·2·1 = 6, 4! = 4·3·2·1 = 24, 5! = 5·4·3·2·1 = 120, etc.

Method 2:

The Taylor series for the exponential function ex at a = 0 is

1 + frac{x^1}{1!} + frac{x^2}{2!} + frac{x^3}{3!} + frac{x^4}{4!} + frac{x^5}{5!}+ cdots = 1 + x + frac{x^2}{2} + frac{x^3}{6} + frac{x^4}{24} + frac{x^5}{120} + cdots! = sum_{n=0}^infty frac{x^n}{n!}.

If x is one, the value of e is equal to 1 + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + 1/6! + 1/7! + ... (etc)

Another way of expressing this series is the formula: e=n=0∞1n!

The first few terms add up to: 1 + 1 + 1/2 + 1/6 + 1/24 + 1/120 = 2.718055556

Method 3:

Recently, new formulae have been developed by Brothers (2004) makes the calculation of e very efficient. e=n=0∞2n+22n+1!