Module 6 : Precalculus

605 : Sequences & Series

S E T U P

In this project we will use the following command packages. Type and execute this line before begining the project below. If you re-enter the worksheet for this project, be sure to re-execute this statement before jumping to any point in the worksheet.

> restart; with(plots): 

Warning, the name changecoords has been redefined

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A. Sequences

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Using the $ operator, we can easily create sequence of numbers in Maple.

Prototype for the sequence, $ operator, a range for k.

> 2*k^2 + 11 $ k = 1..10;

13, 19, 29, 43, 61, 83, 109, 139, 173, 211

Using this method, we can create many types of sequences.

arithmetic sequences

> 2*j + 1 $ j = 1..16;

3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29,...

> 5*k + 17 $ k = 1..16;

22, 27, 32, 37, 42, 47, 52, 57, 62, 67, 72, 77, 82,...

geometric sequences

> 3^k $ k = 1..16;

3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 1...

> (4/5)^k $ k = 1..16;

4/5, 16/25, 64/125, 256/625, 1024/3125, 4096/15625,...

polynomial sequences

> k^2 + 5*k - 3 $ k = 1..16;

3, 11, 21, 33, 47, 63, 81, 101, 123, 147, 173, 201,...

trigonometric sequences

> sin(k*Pi/2) $ k = 1..16;

1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0

other sequences

> 1 + (-1)^k $ k = 1..16;

0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2

> (k! + 1) / (K^3 + k) $ k = 1..16;

2*1/(K^3+1), 3*1/(K^3+2), 7*1/(K^3+3), 25*1/(K^3+4)...
2*1/(K^3+1), 3*1/(K^3+2), 7*1/(K^3+3), 25*1/(K^3+4)...

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B. Series & Sigma Notation

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When we add up a sequence of numbers the result is a sum or series. To express a sum in Maple, we can use the Sum command (with a capital S). This command write the sum in sigma notation, but not compute its value. There are two different ways of computing its value using the sum (with a lower case s), and the value command immediately after the sum.

> Sum( 3*k + 7, k = 1..n) = sum( k , k = 1..n);

Sum(3*k+7,k = 1 .. n) = 1/2*(n+1)^2-1/2*n-1/2

> Sum( 3*k + 7, k = 1..200); value(%);

Sum(3*k+7,k = 1 .. 200)

61700

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C. Rules of Series

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The distributive property for sums looks like this . We can verify that this rule is valid by computing the left and right sides of this equation, and then see that the results are the same. Let k = 4k + 9, and let c = 13.

> Sum( 13* (4*k + 9), k = 1..200); Left := value(%);
Sum( 4*k + 9, k = 1..200); Right := value(%);

Sum(52*k+117,k = 1 .. 200)

Left := 1068600

Sum(4*k+9,k = 1 .. 200)

Right := 82200

We can see for ourselves that the left and right sides are equal value, but we can ask Maple to verify that the left and right values are the same.

> testeq( Left = Right);

false

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D. Series Formulas

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There are formulas to compute the sum of consecutive integers, squares, cubes, etc. You can find many of these formulae in your textbook or a reference book, or let Maple find the formula for you. To get an attractive formula, we will compute the sum, simplify the result, and factor that result.

You might know this formula already. This is the sum of integers : 1 + 2 + 3 + ... + n

> Sum( k, k = 1..n); value(%); simplify(%); factor(%);

Sum(k,k = 1 .. n)

1/2*(n+1)^2-1/2*n-1/2

1/2*n^2+1/2*n

1/2*n*(n+1)

This one is more obscure, the sum of 8th powers of integers : 1^8 + 2^8 + 3^8 + ... + n^8

> Sum( k^8, k = 1..n); value(%); simplify(%); factor(%);

Sum(k^8,k = 1 .. n)

1/9*(n+1)^9-1/2*(n+1)^8+2/3*(n+1)^7-7/15*(n+1)^5+2/...

2/9*n^3-7/15*n^5+2/3*n^7+1/2*n^8+1/9*n^9-1/30*n

1/90*n*(2*n+1)*(n+1)*(5*n^6+15*n^5+5*n^4-15*n^3-n^2...

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E. Infinite Series - Convergence & Divergence

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When you add up an infinite number of numbers, it is very like to get infinity as the result.

> Sum( 3*k - 4, k = 1..infinity); % = value(%);

Sum(3*k-4,k = 1 .. infinity)

Sum(3*k-4,k = 1 .. infinity) = infinity

However, the result is not necessarily infinite. If the numbers get small quickly enough, the sum may be a finite number.

> Sum( (4/5)^k, k = 1..infinity); % = value(%);

Sum((4/5)^k,k = 1 .. infinity)

Sum((4/5)^k,k = 1 .. infinity) = 4

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F. Double Sums

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A more complicated situation is double sum where there is a sum within a sum : . Notice that the limit of the inner sum, m, is the index of the outer sum. The inner sum is adding one number, then two numbers, then three, etc. The outer sum is adding all of these together.

Although this is much more complicated, we can get a formula for this too. We simply nest one Sum command within another.

> Sum( Sum(k^2, k = 1..m), m = 1..N); factor( simplify( value(%)));

Sum(Sum(k^2,k = 1 .. m),m = 1 .. N)

1/12*N*(N+2)*(N+1)^2

> subs( N = 100, %);

8670850

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