Sequences
• Informally, a sequence is a set of elements
written in a row.
– This concept is represented in CS using onedimensional 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 science
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
bound
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 n
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 = 24
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
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! is
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 Products
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 substitution
Dummy Variables in a Loop
• The index is a dummy variable
– Example:
for i := 1 to n
print a[i]
next i
and
for k := 2 to n+1
print a[k-1]
next k
and
for j := 0 to n-1
print a[j+1]
next j
• 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
Recursive Summation Algorithm
• The following is the recursive form of
summation s:=a[1]
for k:= 2 to n
s:= s + a[k]
next k
• This is equivalent to an other recursive
implementation, which doesn’t begin with an
element of the array s:=0
for k:= 1 to n
s:= s + a[k]
next k
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 7
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 find
Formula: a[n] = a[1] + (n-1)*d
To find d, subtract any term a[k] from the next term a[k+1]
• Geometric sequences:
–
–
–
–
Ratio, r, is constant
Initial 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 sequence
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 geometric
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
converge
• If a series proves to converge, sum a large number of elements and
approximate the value.
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)