MATH1020 Calculus II	Maple Lab 6
Sections 1-4	Due: Tuesday, October 26, 1999.
http://www.math.rpi.edu/~mitchj/math1020

In this lab, we examine the behavior of two series and determine their convergence. Be sure to follow the lab guidelines as described on the class handouts.

1.

Let us consider the geometric series given by a_n=1/n^6/5. We are interested in the infinite sum s of the series

$$\sum_{n=1}^\infty {1\over n^{6/5}}$$

This is a slowly converging series. We will investigate how slowly by using Maple. The partial sums of the series can be found by using the Maple command:

> a:=k->1/(k**(6/5));
> Sn:=n->evalf(sum(a(k),k=1..n));
> Sn(N);

The third command is an ``echo'' that just checks that the partial sums are what we wanted. Note that Maple does not know a simple formula for the partial sums -- if it did, it would use it for Sn(N). In order to find some values for Sn and plot its behavior over a range of values of n type

> Sn(1); Sn(2); Sn(3); Sn(10); Sn(100); Sn(200);
> P:=[seq([n,Sn(n)], n=1..200)]: plot(P);

You can also calculate Sn(1000) and Sn(2000). Anything larger is liable to give Maple problems.

We will now forget that we know the value of the sum, We will try to estimate the sum by using the partial sums and the integral test as discussed in section 11.3 in the text. We want to find bounds on the remainder R_n = s - Sn. We get upper and lower bounds on the remainder R_n=s-Sn:

> RemU := n-> evalf(int(a(k),k=n..infinity));
> RemL := n-> evalf(int(a(k),k=n+1..infinity));

(a)

Find upper and lower bounds for the remainder in the following approximations:

$$s = Sn(10), \;\;\; s = Sn(100), \;\;\; s=Sn(1000)$$

(b)

Give a better estimate for s using Sn(n), RemL(n), and RemU(n) for n=10,100,1000. What is the maximum error of each of these three estimates?

(c)

 What is the smallest value of n so that the error of this better estimate is no larger than 0.001? (Hint: Use solve as in the last lab.)

(d)

Use Maple to find the value of the sum:

> s:=Sn(infinity);

How does this compare with the value of the estimate for the n you found in part 1c?

2.

Let

$$a_n={n 3^n\over n!}$$

and consider the three sequences

$$\{a_n\},\qquad \left\{{a_{n+1}\over a_n}\right\},\qquad \left\{\sum_{k=1}\sp n a_k\right\}$$

Enter and plot these sequences using the Maple commands

> a:=n->n*3**n/n!;
> b:=n->a(n+1)/a(n);
> c:=n->sum(a(k),k=1..n);
> F:=[[n,a(n)] $n=1..20]: plot(F);
> G:=[[n,b(n)] $n=1..20]: plot(G);
> H:=[seq([n,c(n)], n=1..20)]: plot(H);

(I gave different expressions for plotting F and H just to show you different ways to get the same result using Maple. For some examples, one of the methods may not work, so try using the other one.)

For each of these three plots, explain what can be concluded about the convergence of the series $\sum_{n=1}\sp\infty a_n$.