#### Question Details

(solution) Compute the value of e. e is the base of the Natural Logarithms.

see attached for more details: Greatly appreciate your help.

3.4.0 or 3.4.3 python version.

Compute the value of e.

e is the base of the Natural Logarithms. The first few digits of e are:

2.7182818284590452353602874713527

There are many ways of calculating the value of e, but none of them ever give an exact answer,

because e is irrational (not the ratio of two integers).

But it is known to over 1 trillion digits of accuracy! Method 1:

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

1

2

5

10

100

1,000

10,000

100,000 (1 + 1/n)n

2.00000

2.25000

2.48832

2.59374

2.70481

2.71692

2.71815

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 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: =? 1 =0 ! 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. =? ? =0 2+2

(2+1)! Your assignment is write a program that computes the value of e using all three of the methods

described above. Use values of 1, 2, 5, 10, 100, 1000, and 10000 for the first method, and values

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

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