MATH 164, Spring 2001 Due Date: Name(s):
Project 12.2b: Sequences and Series
Objective
To investigate sequences and series using Maple.
Narrative
If you have not already done so, read Sections 12.1­12.2 of the text.
In this project we investigate:
1. the sequence {an} = { 1
2n } and the associated series 
n=1
1
2n ,
2. the sequence {an} = 1
n(n+1) and the associated series 
n=1
1
n(n+1) , and
3. the sequence {an} = n
n+1 and the associated series 
n=1
n
n+1 .
To focus attention on our analysis, we use two procedures: values and graphs. Procedures are fragments
of code which facilitate the repetition of tasks.
Tasks
1. Type the command lines in the left-hand column below into Maple in the order in which they are listed.
These commands define the procedures values and graphs.
> # Project 12.2b: Sequences and Series
> restart: with(plots):
> values := proc(a,N)
local M,n,s,pts,graphs:
M := matrix(N+2,3,(Row,Col)->0):
M[1,1] := n: M[1,2] := a n: M[1,3] := s n:
s := n -> sum(a(i),i=1..n):
for n from 1 to N do
M[n+1,1] := n: M[n+1,2] := evalf(a(n)): M[n+1,3] := evalf(s(n)): od:
n := ´n´:
M[N+2,1] := infinity:
M[N+2,2] := evalf(limit(a(n),n=infinity)):
M[N+2,3] := evalf(limit(s(n),n=infinity)):
RETURN(eval(M)):
end:
> graphs := proc(a,N)
local apts,s,spts,n:
apts := [seq([n,a(n)],n=1..N)]:
s := n -> sum(a(i),i=1..n);
spts := [seq([n,s(n)],n=1..N)]:
plot({apts,spts},x=1..N,style=point);
end:
2. a) Type the command lines in the left-hand column below into Maple in the order in which they are
listed. These commands are aimed at studying the sequence {an} = 1
2n and the associated series

n=1
1
2n .
> values(n -> 1/2^n,10);
> graphs(n -> 1/2^n,10);
b) Repeat part (a) using the sequence {an} = 1
n(n+1) rather than {an} = 1
2n .
c) Repeat part (a) using the sequence {an} = n
n+1 rather than {an} = 1
2n .
At this time, make a hard-copy of your typed input and Maple's responses. Then, ...
3. For each of the (three) parts of Task 2:
a) connect the dots in the sequence {an}, and the dots in the sequence {sn} by hand,
b) label the sequence {an} as "{an}", and the sequence {sn} as "{sn}" by hand, and
c) next to the limit of the sequence {an} write, "The sequence {an} .", filling in the blank
with "converges" or "diverges", and next to the limit of the sequence {sn} write, "The series 
n=1an
.", again filling in the blank with "converges" or "diverges".