MATH 156 LAB 12

Topic 1: Sequences and their limits.

We can define a sequence given by an explicit formula NiMvJiUiYUc2IyUibkctJSJmR0Ym by defining the function NiMtJSJmRzYjJSJ4Rw==. Example: The sequence NiMvJiUiYUc2IyUibkcsJiIjOyIiIiomRilGKikqJkYqRioiIiMhIiJGJ0YqRi8=.

a:=n->16-16*(1/2)^n;

a(1);a(2);a(3);

We can easily make a list of its values with a loop command.

for n from 1 to 20 do evalf(a(n));od;

We see that the numbers get closer and closer to 16. In fact NiMmJSRsaW1HNiMlIm5HNiMmJSJhRzYjJSJuRw===16. We can compute this limit with the Limit command of Maple:

Limit(a(m), m=infinity);

value(%);

Graphically we can see the sequence by creating a list of the points [n, NiMmJSJhRzYjJSJuRw==]:

graph:=[seq([n,a(n)], n=1..10)];

plot(graph, style=point);

The style=point option plots a dot or star at the corresponding point. We can also use the option style=line. This produces:

plot(graph, style=line);

We see that the segments joining the points become eventually almost horizontal at height 16. This is the limit of the sequence. We can also plot the function and see the points of the sequence on it.

gr1:=plot(graph, style=point):

gr2:=plot(a(x), x=0..10):

with(plots); display(gr1, gr2);

Investigate the limit of the sequence NiMmSSJiRzYiNiNJIm5HRiU==NiMpJSMlP0clIm5HNiMtJSVzcXJ0RzYjIiIj. Make a table showing the terms with NiMlIm5H a multiple of 10 and show the sequence graphically.

It seems that the limit is 1.

Investigate the sequence NiMmJSJjRzYjJSJuRw===NiMpLCQqJiIiIkYmIiIkISIiRiglIm5H. Find its limit, make a table and show the first 10 terms of the sequence graphically.

Investigate the sequence NiMmJSJkRzYjJSJuRw===NiMpLSUmRmxvYXRHNiQiIzwhIiIlIm5H. Find its limit, make a table and show the first 10 terms of the sequence graphically.

Example: The Fibonacci sequence. Compute the first 50 terms of the sequence given by NiMvLSUiZkc2IyIiIkYn, NiMvLSUiZkc2IyIiIyIiIg== and the recursion NiM+LSUiZkc2IyUibkcsJi1GJTYjLCZGJyIiIkYsISIiRiwtRiU2IywmRidGLCIiI0YtRiw=

The easiest way to work with sequences defined recursively is to use a loop.

Topic 2: Infinite series and their sums.

Using the sum command we can find the partial sums of various series and then their sum: Example: The series NiMtJSRzdW1HNiQpKiYiIiJGKCIiIyEiIiUibkcvRis7RiglKWluZmluaXR5Rw==. Its partial sums NiMmJSJzRzYjJSJORw== are given by NiMmJSJzRzYjJSJORw===NiMtJSRzdW1HNiQpKiYiIiJGKCIiIyEiIiUibkcvRis7RiglIk5H. We can use a loop to calculate NiMmJSJzRzYjJSJORw== for NiMvJSJORyIiIg== .. 30.

restart;

s:=N->sum((1/2)^n,n = 1 .. N);

for N from 1 to 30 do evalf(s(N));od;

It seems that the limit of the partial sums is 1. This is the sum of the infinite series. Here is the graph of the partial sums:

graph:=[seq([m, s(m)], m=1..20)];

plot(graph, style=point);

We can ask Maple whether it can calculate a formula for the partial sums. The answer is yes.

restart;s(N):=sum((1/2)^n, n=1..N);

Limit(s(N), N=infinity);

value(%);

In fact we saw the formula NiMtJSRzdW1HNiQqJiUiYUciIiIpJSJyRyUibkdGKC9GKzsiIiElIk5H=NiMqKCUiYUciIiIsJkYlRiUpJSJyRywmJSJOR0YlRiVGJSEiIkYlLCZGJUYlRihGK0Yr in class. Verify the formula for NiMvJSJORyIiIg== .. 20 with NiMvJSJhRyIiJg== and NiMvJSJyRyomIiIjIiIiIiIkISIi.

Explain why NiMtJSRzdW1HNiQqJiIiJiIiIikqJiIiI0YoIiIkISIiJSJuR0YoL0YuOyIiISUpaW5maW5pdHlH=15.

Compute the sum of the series NiMtJSRzdW1HNiQqJiIiIkYnKiYlIm5HRicsJkYpRiciIiNGJ0YnISIiL0YpO0YnJSlpbmZpbml0eUc=. This is a telescoping series. To see where its sum comes from, compute the partial sums NiMmJSJzRzYjJSJORw== and use partial fractions to NiMqJiIiIkYkKiYlIm5HRiQsJkYmRiQiIiNGJEYkISIi to see the cancellation.