Sequences - PowerPoint Presentation

Slide1

Sequences

Informally, a sequence is a set of elements written in a row.

This concept is represented in CS using one-dimensional arrays

The goal of mathematics in general is to identify, prove, and utilize patterns

The most common way to do this is to use sequences to represent outputs

The most efficient way to use sequences is through computer scienceSlide2

Sequence Operations

There are two basic operations that can be performed on sequences

Summation, denoted by ∑ and multiplication, denoted by ∏

The sum of a finite number of elements in a sequence is a partial sum

When written on paper, values would be included on the top and bottom of each notation. The top value represents the highest element in the sequence to be evaluated and the bottom represents the lowest value.

These are referred to as the upper- and lower- bounds. Each element between the bounds inclusive is evaluated. The lower bound must be less than or equal to the upper boundSlide3

Series

The summation of a sequence is particularly useful in mathematics

A series is the written result of the summation of a sequence

EX: 1 + 1/2 + 1/3

+ 1/4+...+1/n

If an infinite summation is performed, the series is referred to as an infinite series

If a series is finite, it can be represented as a simple summation, or a partial sum to

nSlide4

Examples

Summation:

If n = 3, the summation of 2 raised to k is 2^1 + 2^2 + 2^3 = 14

The general summation formula is

Products:

If n = 3, m = 1 and the formula is (k+1) then the answer is (1+1)(2+1)(3+1) = 2*3*4 = 24Slide5

Recursive Form of Summation

m and n are any integers, n > m

This definition lets you separate out the final term or condense a complex statement into a smaller one.

EX: a telescoping sum

By breaking the term into two parts, you can cancel and simplify the summation formula

Everything from 1 to 1/(n+1) cancels out, so the final formula for the sum is Slide6

Recursive Form of Products

m and n are any integers, n > m

A common use of recursive product form is computing the factorial,

n!

, which is the product of

n

and each integer less than

n

and greater than 1 inclusive. Note that 0! = 1 The recursive definition of n! isSlide7

Properties of Summations and Products

Assume and are real number sequences and

c

is a real number and

n ≥ m

--Addition of Sums

-- Generalized Distributive Law

--Multiplication of ProductsSlide8

Dummy Variables

All of the variables (k, m, n) in the two common notations are generic because they have no special meaning except for their context

Say you have (k-1) in the notation and you want to replace it with j. Make j = k-1, then j+1 = k. Everywhere you see a k, put j+1 in its place. This is a variable substitutionSlide9

Dummy Variables in a Loop

The index is a dummy variable

Example: and

and

The above three for-next loops yield the same output.

i

, k, and j are dummy variables because they can be substituted anywhere to obtain the same result

for

i

:= 1 to n print a[i]next

ifor j

:= 0 to n-1 print a[j+1]next j

for k := 2 to n+

1

print a[k-1]

next kSlide10

Recursive Summation Algorithm

The following is the recursive form of summation

This is equivalent to an other recursive implementation, which doesn’t begin with an element of the array

s:=a[1]

for k:= 2 to

n

s:= s + a[k]

next k

s:=0

for k:= 1 to n

s:= s + a[k]next k Slide11

Common Sequences

Two very common styles of sequences are arithmetic and geometric sequences

An arithmetic sequence is one where the difference between terms is constant

A geometric sequence is one where the ratio between terms is constant

Other, more complicated sequences exist too

In order to find them, you have to be able to recognize common patterns like recurrence, which is discussed in section 7Slide12

Explicit Formulas

Arithmetic sequences:

Difference,

d

, is constant

Initial condition,

a

[1], is the first term and is given

a[n] is the term we want to findFormula: a[n] = a[1] + (n-1)*dTo find d, subtract any term a[k] from the next term a[k+1]Geometric sequences:Ratio, r

, is constantInitial condition a[1]Formula: a[n] = a[1]*r^(n-1)To find r, divide any term a[k] by the next term a[k+1]By plugging in the number of the element you want for

n, you can find any term of the sequenceSlide13

Sigma Notation for a Series

A sigma, as earlier denoted, means summation

Sigma notation represents the series up to a certain element

n

The sigma notation is equivalent to the summation formula for a sequence

Two formulas to know for now are the summations of arithmetic and geometric sequences

for arithmetic for geometricSlide14

Converging Series

In certain, special kinds of sequences, there is a property called convergence

Consider the geometric sequence:

Each term of a geometric sequence for

n

≥ 1 is closer to 0 than the last if the absolute value of the ration is less than 1

Interestingly, if you add each of these values together all the way to infinity, you will achieve a finite value

EX: 10^-x = .1, .01, .001, .0001… when you add them, you get .1111111… which is 1/9

Not all series with smaller forward values converge, so a test has been developed

As n approaches positive infinity (just use a large value like 100. If it doesn’t work, use 1000), divide the term a[n+1] by a[n] If p<1, the series will converge. If p=1, the series may converge, but probably won’t; in this case, use a different value for n. If n > 1, it won’t convergeIf a series proves to converge, sum a large number of elements and approximate the value.Slide15

Finding Infinite Sums

Once a series is proven to converge, an effort can be made to find its infinite sum

To calculate the exact value of an infinite sum requires mathematical analysis, such as calculus, in most situations

For this level of discrete mathematics, approximations of infinite sums will do just fine

To approximate, use a for loop to add successive terms (up to a high number, like 10,000)