Graphing Sequences and Series

Syntax for Doing It Quickly

>	with(plots):

Warning, the name changecoords has been redefined

These commands will accomplish the goal a bit quicker than some of the other worksheets I have put online.

I will begin with a sequence.

>	pointplot({seq([n,sin(n/10)],n=0..314)},color=red,symbol=circle,symbolsize=12,labels=[n,an]);

Here is a geometric sequence.

>	pointplot({seq([n,(4/5)^n],n=0..30)},color=red,symbol=circle,symbolsize=12,labels=[n,an]);

I will ask Maple to print the first few terms in the sequence graphed above.

>	seq((4/5)^n,n=0..10);

Here are some decimal approximations.

>	evalf(seq((4/5)^n,n=0..10));

Here is a partial sums graph of the geometric series associated with the sequence directly above.

>	pointplot({seq([n,sum((4/5)^i,i=0..n)],n=0..30)},color=red,symbol=circle,symbolsize=12,labels=[n,S]);

Let's ask Maple to compute a finite and the infinite sum related to the geometric series above.

>	sum((4/5)^i,i=0..30);

>

evalf(%);

>	sum((4/5)^i,i=0..infinity);

Using a capital S will result in a display that can then be evaluated using the "value" command.

>	Sum((4/5)^i,i=0..infinity);

>

value(%);

We can use the capital S in the sequence command but following with the value command does not result in each term being displayed separately.

>	Seq((4/5)^n,n=0..10);

>

value(%);

Observe that the formula developed in class works.

>	Sum(10*(2/5)^i,i=0..infinity);

>

value(%);

>	10/(1-2/5);

Here is a graph of partial sums related to the geometric series directly above.

>	pointplot({seq([n,sum(10*(2/5)^i,i=0..n)],n=0..30)},color=red,symbol=circle,symbolsize=12,labels=[n,S]);

The partial sum formula for a geometric series summing from 0 also works.

>	Sum(10*(2/5)^i,i=0..30);

>

value(%);

>

evalf(%);

>	(10-10*(2/5)^31)/(1-2/5);

>

evalf(%);

I will conclude this worksheet with a telescoping series.

>	seq(4/(n^2-1),n=2..12);

>	seq(2/(n-1),n=2..12)-seq(2/(n+1),n=2..12);

>	seq(2/(n-1),n=2..14);

>	seq(-2/(n+1),n=2..12);

>	Sum(4/(n^2-1),n=2..12);

>

value(%);

>

evalf(%);

>	evalf(sum(4/(n^2-1),n=2..100));

>	evalf(sum(4/(n^2-1),n=2..1000));

>	Sum(4/(n^2-1),n=2..infinity);

>

>

value(%);

The idea is that everything will "subtract out" except for 2 + 1.

Here is a graph of partial sums.

>	pointplot({seq([n,sum(4/(i^2-1),i=2..n)],n=2..60)},color=red,symbol=circle,symbolsize=12,labels=[n,S]);

>