Calculus B - Lab 8 

TA: Patrick C. Rowe 

Lab adapted from material by Ray Viglione 

Today we look at sequences and series.  A sequence is just a list of numbers; as such we can just define them as functions on Maple.  To define a[n] = 2*n^2/(n^2+1) , we use typical function notation: 

>	a:=n->2*n^2/(n^2+1);

 

Maple can list elements of the sequence.  For example, to list out the tenth thru twentieth terms, type: 

>	seq(a(n),n=10..20);

 

We can plot the sequence.  To plot the first 100 terms of the above sequence: 

>	plot( [ seq([n,a(n)],n=1..100) ] , style=point, symbol=cross);

 

(Other symbol options are diamond, box, circle, and point.)  The above sequence appears to have a limit, namely 2.  We evaluate it in the usual way. 

>	limit(a(n),n=infinity);

 

We can also implement sums and series using the "sum" command.  Using the same a[n] as above, let's find     . 

>	Sum(a(n),n=1..200);

 

>

value(%);

 

>

evalf(%);

 

We don't have to use a capital "s" in our command. The capital "S" is the inert form. 

>	sum(ln(n),n=4..7);

 

>

evalf(%);

 

We can sum infinite series.  If Maple returns the infinity symbol, then the series diverges.  You may recognize the first series as a simple geometric series.  The second series diverges because, as we saw, the terms don't go to zero. 

>	sum(1/3^(n-1),n=1..infinity);

 

>	Sum(a(n),n=1..infinity);

 

>

value(%);

 

>