Download 1. Euler`s number - Euler`s number `e` is number toward which

1.
Euler's number
 Euler's number 'e' is number toward which function f(x) goes but never reaches it when x goes into infinity:
 1
f ( x )  1  
 x
x
 1
e  lim 1  
x  
x
(1.1)
x
 Function f(x) as defined with (1.1) is shown on following figure
f(x)=(1+1/x)x
15
10
5
f(x)
e  2.718281828
0
-5
-10
-15
-5
-4
-3
-2
-1
0
1
2
3
4
5
x
Figure 1.1.
Euler's number.
 Figure 1.1. was created using following MATLAB code:
clear;
%DEFINE f(x).
x =[-5:0.01:5];
fx=(1+1./x).^x;
%PREPARE FIGURE.
figure(1); clf; grid on; hold on; axis([-5 5 -15 15]);
title('f(x)=(1+1/x)^x','FontSize',14); xlabel('x','FontSize',14);
ylabel('f(x)','FontSize',14);
line([-5 5],[0
0],'Color','k','LineWidth',3);
line([0 0],[-15 15],'Color','k','LineWidth',3);
%DRAW e.
e = 2.718281828;
text(1.05,3.5,'e \approx 2.718281828','FontSize',14)
line([-5 5],[e e],'Color','r','LineWidth',2);
%DRAW f(x).
plot(x,fx,'LineWidth',2);
1.1.
Expressing Euler's number as the sum of infinite series
 In this chapter we will show that Euler's number can be expressed as the sum of infinite series like this [1]:
n

1 1
1
 1
e = lim 1   = 1  1     = 
n 
n
2! 3!
n 0 n!
 We will start our proof with binomial theorem which looks like this:
1  x n  1  nx  n n  1 x 2  n n  1n  2 x 3    x n
2!
(1.2)
3!
 Substituting:
x
1
n
we get:
 1
1  
 n
n
2
= 1 n
3
1 n n  1  1 
n n  1n  2   1 
1

  
    
n
2!  n 
3!
n
n
= 11
n
1 n n  1 1 n n  1n  2 
1

 n
2
3
2! n
3!
n
n
 We are only interested what happens to this series when n goes toward infinity and then (1.3) can be rewritten like:
 1
lim 1  
n 
n
n
= 11
1 n2 1 n3
1

 n
2! n 2 3! n 3
n
= 11
1 1
  0
2! 3!
= 11
1 1
 
2! 3!

=
1
 n!
n 0
1.2.
Propertie of Euler's number
 Cut a number into equal parts and then multiply those parts together. [3]
 How large should each part be, so that when we multiply them together they make the biggest possible number?
 The answer is to make the parts as close to e as possible.
10 cut into 3 equal parts is 3.3
10 cut into 4 equal parts is 2.5
10 cut into 5 equal parts is 2



3.3 3 = 37.037...
2.54 = 39.0625
25 = 32
(1.3)
2.
References
[1]
http://galileo.phys.virginia.edu/classes/152.mf1i.spring02/Exponential_Function.htm
[2]
http://planetmath.org/encyclopedia/DerivativeOfExponentialFunction.html
[3]
http://www.mathsisfun.com/numbers/e-eulers-number.html
[4]
http://en.wikipedia.org/wiki/E_(mathematical_constant)