Numerical Investigations of Sequences and
Series
Tom Gilsdorf, University of North Dakota
Objectives of Assignment
1.
2.
3.
I.
To learn how to generate sequences using X(plore).
To learn how to evaluate partial sums of series using X(plore).
To practice guessing whether sequences and series converge or diverge,
and estimating the value of the limit of convergent sequences and series.
Using X(plore) to Generate a Sequence
X(plore) uses the list command to generate a list. To generate an = n2 for n = 1 to n =
5, type and press E
A = list(n^2, n = 1 to 5 step 1).
Now change to n = 1 to 100 and press E.
Now change to n = 1 to 8 step 2 and press E.
1
. (To check your work, a19 = 3.99723
n2
to five decimal places, for example.)
Example 1.
Generate the sequence an = 4 –
Example 2.
If ε = 0.0001, find the smallest N such that a n − 4 < ε, for all n ≥ N
1
where an = 4 – 2 . Ans: N = 101
n
II.
Using X(plore) to Evaluate a Partial Sum of a Series
First use list to define a sequence. Then use sum to add the terms of the sequence.
25
For example, to find
∑n
3
,
n =1
Type:
A = list (n^3, n = 1 to 25 step 1) and press E, then
37
Type:
sum(A) and press E.
You should get ANS = 105,625.
Example 3.
(i)
Define a list for the sequence a n =
(ii)
Use the sum command to estimate
n
.
n + 19
3
4000
∑a
n
.
n=1
(iii)
Try to find the sum S of the series. Look for an N such
that S − S N < 0001
.
.
Ans: (ii) 0.448519
38