Calculus & Analytic Geometry II, MTH 169  

(Instructor: M. USMAN) 

Lab#2 

Objectives: In this lab we will apply Integral Test  and Direct Comparison Test to check the convergence of series. 

Integral Test 

Estimates of sum from integral test: Let f be a positive, continuous and decreasing function for  

If  

Example. Use Maple to evaluate the infinite sum   . 

>	sum('1/n^2', 'n'=1..infinity);

 

>

evalf(%);

 

Example. Use Maple to evaluate the infinite sum   . 

>	sum('(ln(n+1))^2/(n^2)', 'n'=1..infinity);

 

>

evalf(%);

 

Maple did not evaluate the sum, we can use the Integral Test to convince ourselves that this series converges. Lets plot the function to see if it is positive, decreasing and continuous on [1,infinity) 

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

 

>	f:=x->(ln(x+1))^2/x^2;

 

(1)

 

>	int(f,1..infinity);

 

>

evalf(%);

 

Since the integral converges, the series does too. Remember that the series and the integral are not equal. We can sketch f(x) and the inscribed rectangles (with bases having width 1) which represent the terms of the series: 

>	with(student):

 

>	f:=x->(ln(x+1))^2/x^2;

 

(2)

 

>	rightbox(f(x), x=1..51,50);

 

One way to approximate an infinite sum is to use a partial sum  -- a sum of a finite number of terms.  Suppose we want to use the first 50 terms of this series as an approximation to the sum of the infinite series. 

>	s50:=evalf(sum('f(n)','n'=1..50));

 

An estimate for the errror (discussed in class) is that: 

                        < <     

We can have Maple compute these integrals for us. 

>	leb:=evalf(int(f, 51..infinity));

 

>	ueb:=evalf(int(f,50..infinity));

 

Since the error is between leb and ueb, the sum S of the whole series lies in the interval  

[]. An improved estimate for S is the midpoint of this interval: 

>	ImpS:=s50+(leb+ueb)/2;

 

The error is now at most: 

>	(ueb-leb)/2;

 

Comparison Test: 

Let  

1. If  

2. If  

Example: Use the Direct Comparison Test to show that the following series converges 

Solution: 

= 4  

with p=3>1 so it converges and thus  

Thus