Write a Java Program to 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 eis 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:
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n
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(1 + 1/n)n
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1
|
2.00000
|
|
2
|
2.25000
|
|
5
|
2.48832
|
|
10
|
2.59374
|
|
100
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2.70481
|
|
1,000
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2.71692
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|
10,000
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2.71815
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100,000
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2.71827
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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 zero, 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:
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) which make the calculation of e very efficient.
Your assignment is write a program that computes the value of e using all three of the methods described above for the values of n = 1,2,3,4,5,6,7,8,9,10, and 20. Your program must print out the value of e and the difference with the value of in Math. Make certain that the outputs are labeled as to which method is being used and what value of n was used.