Calculus & Analytic Geometry II, MTH 169  

(Instructor: M. USMAN) 

Lab#3 

Sequence and Series 

To create a MAPLE sequence, use the command: 

>	a := seq(n/(n+1), n = 1 .. 10);

 

(1)

 

We can access the terms of the sequence using a[1], a[2], ... 

>	a[3];a[10];

 

 

	(2)

 

To plot a sequence, we provide a "list" of values  [in square brackets] for MAPLE to plot: 

>	seq_list:=[seq([n,n/(n+1)], n = 1 .. 30)];

 

(3)

 

>	plot(seq_list,style=POINT);

 

What is your guess for the limit? may be one? Lets compute its limit. 

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

 

(4)

 

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

 

(5)

 

>

 

Convergence of  a Series and Integral Test 

Recall that series converges if the sequence of its partial sums converges. 

Consider the series  

>	sn_partialSum:=n -> sum(1/(k^2),k=1..n);

 

(6)

 

>	seqpsum_list:=[seq([n,sum(1/k^2, k = 1 .. n)], n = 1 .. 50)]:

 

>	plot(seqpsum_list,style=POINT);

 

>	evalf(sum(1/k^2, k = 1 .. infinity));

 

(7)

 

Many infinite series are not geometric, and so a large portion of this course is spent developing tests for infinite series which tell us if the series converges or diverges. When geometric series converge, we can always find the sum of the series: S = a/1-r.  For most series, however, we will only be able to tell if the series converges or diverges . Finding the sum of the series is often just too difficult by hand.Integral Test:If f(x) is a continuous, positive and decreasing function on [1, ∞) and = f(n) for each term in the positive termed series ,then converges if and only if  converges. 

Look at the example First check that the corresponding function f(x) =  is continuous, positive and decreasing on [1,∞) by graphing f(x). 

>	plot(1/x^2,x=1..10);

 

Then check if the integral from 1 to ∞ converges or not: 

>

 

(8)

 

Since the improper integral     is convergent, the infinite series is also convergent. 

Your turn: (Turn in the solution of this problem on Monday, March 16th) 

Q#1 Plot first 50 terms of sequence of partial sums of the series . Use the integral test to verify that the series converges.  Then, find the maximum error if the first 20 terms of the series are used to approximate the actual sum.