MAT - NYB 01 H2001

Rates of growth

> Limit(n^4/2^n, n=infinity) =limit(n^4/2^n, n=infinity); exponential increases faster than any power function

> Limit(n^10/10^n, n=infinity)=limit(n^10/10^n, n=infinity);

> Limit(2^n/n!, n=infinity)=limit(2^n/n!, n=infinity); factorial increases faster than any exponential

> Limit(n^n/n!, n=infinity)=limit(n^n/n!, n=infinity); the "coupled exponential" increases faster than the factorial

> Limit(n!/n^n, n=infinity)=limit(n!/n^n, n=infinity);

>

Exponential types eventually increase faster than any fixed power function, but this need not happen in the short run:

> plot( [x^4, 2^x], x = 0..3,y = 0..10,color = [red,blue],thickness = [1,2],
title = "x^4 and 2^x for small values of x");

Where do these curves meet at first?

> fsolve(x^4 = 2^x,x = 1.. 1.3);

S oat first the power function overtakes the exponential at approx x = 1.2396. But it is guaranteed to lose out to the exponential later on.

How large must x be before ?

> plot({x^4/2^x, 1}, x = 1..20);

> fsolve(x^4 = 2^x, x = 15..18);

> (16^4 = 2^16);

Conclusion: the exponential function y = catches up with the power function y = at x = 16.

> plot( [x^4, 2^x], x = 1..17,color = [red,blue],thickness = [1,2],
title = "x^4 and 2^x for larger values of x");

Monotone and Bounded ===> Convergent

Here we demonstrate that the sequence is increasing and bounded above, hence convergent.

The smallest upper bound = the limit = e.

> plot([(1+1/n)^n,exp(1), 2.73], n = 10..200, style= [point,line,line], color = [red,blue,magenta], thickness = [1,3,2]);

> Limit( (1+1/n)^n, n=infinity): % = value(%);

> a := n->(1+1/n)^n; Maple syrup: defining a sequence { } as a function of n

> a(n);

Show that this sequence is increasing:

> ratio := a(n+1)/a(n); we will demonstrate that ratio > 1 for all n [not a rigorous proof, just a visual demo.]

> simplify(ratio);

Plot ratio and the constant function y = 1 o the same graph, on various n- intervals

> plot([ratio, 1],n = 10..100,style=[point, line]);

> plot([ratio, 1],n = 100..200,style=[point, line]);

> plot([ratio, 1],n = 500..1000,style=[point, line]);

This last picture is no longer accurate (those horizontal stretches are due to hardware limitations - pixel resolution)

but it shows quite emphatically that ratio > 1.

But this means a(n+1) > a(n) , i.e. the sequence {a(n) } is (strictly) increasing.