SequencesLimits.html

Sequences

some examples

> with(plots):

Warning, the name changecoords has been redefined

Consider the sequence ${2*(2/3)^n}$ .

> points:=array(1..40);

> for i from 1 to 40 do

> points[i]:=(i,2*(2/3)^i):

> end do:

> pointss:=convert(points,listlist):

> pointplot(pointss,symbol=circle,symbolsize=12,color=red);

> Limit(2*(2/3)^n,n=infinity);

> value(%);

The sequence converges to 0.

Consider the sequence ${(12*n^3-4*n+8)/(4*n^3+16*n^2+5)}$ .

> points2:=array(1..40);

> for i from 10 by 10 to 400 do

> points2[i/10]:=(i,(12*i^3-4*i+8)/(4*i^3+16*i^2+5)):

> end do:

> points2s:=convert(points2,listlist):

> pointsgraph:=pointplot(points2s,symbol=circle,symbolsize=12,color=red):

> upperbound:=plot(3,x=0..400,y=2..3,thickness=2,color=blue):

> display(pointsgraph,upperbound);

> Limit((12*n^3-4*n+8)/(4*n^3+16*n^2+5),n=infinity);

> value(%);

The sequence is bounded above by 3 and converges to 3.

Let's consider the sequence .

> points3:=array(1..400);

> for i from 1 to 400 do

> points3[i]:=(i,cos(sqrt(i))):

> end do:

> points3s:=convert(points3,listlist):

> pointplot(points3s,symbol=circle,symbolsize=12,color=red);

> Limit(cos(sqrt(n)),n=infinity);

> value(%);

The sequence does not converge. It fluctuates between the values of -1 and 1.

This time let's look at .

> points4:=array(1..400);

> for i from 1 to 400 do

> points4[i]:=(i,cos(sqrt(i))/i):

> end do:

> points4s:=convert(points4,listlist):

> pointplot(points4s,symbol=circle,symbolsize=12,color=red);

> Limit(cos(sqrt(n))/n,n=infinity);

> value(%);

The sequence converges to 0.

The next sequence I will look at is ${n^3/n!}$ .

> points5:=array(1..40);

> for i from 1 to 40 do

> points5[i]:=(i,i^3/i!):

> end do:

> points5s:=convert(points5,listlist):

> pointplot(points5s,symbol=circle,symbolsize=12,color=red);

> Limit(n^3/n!,n=infinity);

> value(%);

The sequence converges to 0.

What about ${n^10/n!}$ ?

> points6:=array(1..40);

> for i from 1 to 40 do

> points6[i]:=(i,i^10/i!):

> end do:

> points6s:=convert(points6,listlist):

> pointplot(points6s,symbol=circle,symbolsize=12,color=red);

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

> value(%);

The sequence still converges to 0.

What about ${n^n/n!}$ ?

> points7:=array(1..10);

> for i from 1 to 10 do

> points7[i]:=(i,i^i/i!):

> end do:

> points7s:=convert(points7,listlist):

> pointplot(points7s,symbol=circle,symbolsize=12,color=red);

> points7:=array(1..20);

> for i from 1 to 20 do

> points7[i]:=(i,i^i/i!):

> end do:

> points7s:=convert(points7,listlist):

> pointplot(points7s,symbol=circle,symbolsize=12,color=red);

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

> value(%);

This sequence diverges.

If you invest $10,000 at an annual rate of interest of 5.5% compounded monthly, then the value of your investment each month forms a sequence. The sequence is represented by

{10000*(1+0.055/12)^n} where n represents the number of months the money was invested. Let's look at how much the value of the investment will grow if someone invested the money at age 25 and left it alone until age 65 (40 years, i.e., 480 months).

> points8:=array(1..120);

> for i from 4 by 4 to 480 do

> points8[i/4]:=(i,10000*(1+0.055/12)^i):

> end do:

> points8s:=convert(points8,listlist):

> pointplot(points8s,symbol=circle,symbolsize=12,color=red);

> points8[120];

So after 40 years the value of the investment would be $89,797.63. If you could live forever and left the money invested at 5.5% compounded monthly, what would be the limiting value of the investment?

> 10000*Limit((1+0.055/12)^n,n=infinity);

> value(%);

The sequence diverges and the investment is approaching an infinite value.

In how many years would the investment reach a value exceeding one million dollars?

> fsolve(10000*(1+0.055/12)^n=1000000,n);

> 1008/12;

It would take 1008 months, i.e., 84 years, for the value of the investment to exceed one million dollars.

The last sequence I'll look at here is ${n^5/exp(n)}$ .

> points9:=array(1..60);

> for i from 1 to 60 do

> points9[i]:=(i,i^5/exp(i)):

> end do:

> points9s:=convert(points9,listlist):

> pointplot(points9s,symbol=circle,symbolsize=12,color=red);

> Limit(n^5/exp(n),n=infinity);

> value(%);

This sequence converges to 0. Note that we could have established this convergence using the corresponding real valued function and applying L'Hopital's Rule a few times.