Exponential and Logarithmic Functions

Contents

What is this number e, anyway?
Let's plot both curves
From the plot, you can see that as n grows, $(1+1/n)^n \rightarrow e$

What is this number e, anyway?

Let's examine the curves y = (1+1/n)^n and y =e.

%  The set of input "x" values is the array n
	n = .01:100;
	y = @(n) (1 + 1./n).^n;

% Let's also examine the line  y = e.
	e = exp(1);
	yy = e*ones(size(n));

Let's plot both curves

	figure; hold on;
	title(['\fontsize{16}Relationship between y = (1+1/n)^n and the natural number e']);
	b = plot(n,y(n),'-k',n,yy,'-r');
	legend('\fontsize{16}y = (1+1/n)^n','\fontsize{16}y=e','Location','SouthEast')
	xlabel('n');
	ylabel('(1+1/n)^n');

From the plot, you can see that as n grows, $(1+1/n)^n \rightarrow e$