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Sequences. Rosen 6 th ed., §2.4. Sequences. A sequence represents an ordered list of elements. e.g., { a n } = 1, 1/2, 1/3, 1/4, …

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sequences

Sequences

Rosen 6th ed., §2.4

sequences2
Sequences
  • A sequence represents an ordered list of elements.

e.g., {an} = 1, 1/2, 1/3, 1/4, …

  • Formally: A sequence or series {an} is identified with a generating functionf:SA for some subset SN (often S=N or S=N{0}) and for some set A.

e.g., an= f(n) = 1/n.

  • The symbol an denotes f(n), also called term n of the sequence.
example with repetitions
Example with Repetitions
  • Consider the sequence bn = (1)n.
  • {bn}= 1, 1, 1, 1, …
  • {bn} denotes an infinite sequence of 1’s and 1’s, not the 2-element set {1, 1}.
recognizing sequences
Recognizing Sequences
  • Sometimes, you’re given the first few terms of a sequence, and you are asked to find the sequence’s generating function, or a procedure to enumerate the sequence.
recognizing sequences5
Recognizing Sequences
  • Examples: What’s the next number?
    • 1,2,3,4,…
    • 1,3,5,7,9,…
    • 2,3,5,7,11,...

5 (the 5th smallest number >0)

11 (the 6th smallest odd number >0)

13 (the 6th smallest prime number)

the trouble with recognition
The Trouble with Recognition
  • The problem of finding “the” generating function given just an initial subsequence is not well defined.
  • This is because there are infinitely many computable functions that will generate any given initial subsequence.
the trouble with recognition cont d
The Trouble with Recognition (cont’d)
  • We implicitly are supposed to find the simplest such function but, how should we define the simplicity of a function?
    • We might define simplicity as the reciprocal of complexity, but…
    • There are many plausible, competing definitions of complexity, and this is an active research area.
  • So, these questions really have no objective right answer!
summations

Summations

Rosen 6th ed., §2.4

summation notation
Summation Notation
  • Given a series {an}, the summation of {an} from j to k is written and defined as follows:
  • Here, i is called the index of summation.
generalized summations
Generalized Summations
  • For an infinite series, we may write:
  • To sum a function over all members of a set X={x1, x2, …}:
  • Or, if X={x|P(x)}, we may just write:
more summation examples
More Summation Examples
  • An infinite series with a finite sum:
  • Using a predicate to define a set of elements to sum over:
summation manipulations
Summation Manipulations
  • Some handy identities for summations:

(Index shifting.)

more summation manipulations
More Summation Manipulations

(Series splitting.)

(Order reversal.)

(Grouping.)

example impress your friends
Example: Impress Your Friends
  • Boast, “I’m so smart; give me any 2-digit number n, and I’ll add all the numbers from 1 to n in my head in just a few seconds.”
  • I.e., Evaluate the summation:
  • There is a simple closed-form formula for the result, discovered by Euler at age 12!

LeonhardEuler(1707-1783)

euler s trick illustrated
Euler’s Trick, Illustrated
  • Consider the sum (assume n is even):1+2+…+(n/2)+((n/2)+1)+…+(n-1)+n
  • n/2 pairs of elements, each pair summing to n+1, for a total of (n/2)(n+1).

n+1

n+1

n+1

symbolic derivation of trick
Symbolic Derivation of Trick

Suppose n is even, that is, n=2k

order reversal

index shifting

(k=n/2)

concluding euler s derivation
Concluding Euler’s Derivation
  • Also works for odd n (prove this at home).
example geometric series
Example: Geometric Series
  • A geometric series is a series of the form

a, ar, ar2, ar3, …, ark, where a,rR.

  • The sum of such a series is given by:
  • We can reduce this to closed form via clever manipulation of summations...
more series
More Series

Infinite Geometric

more series24
More Series

Geometric series

Arithmetic Series

(Euler’s trick)

Quadratic series

Cubic series

example
Example
  • Evaluate .
    • Use series splitting.
    • Solve for desiredsummation.
    • Apply quadraticseries rule.
    • Evaluate.
nested summations
Nested Summations
  • These have the meaning you’d expect.
  • Note issues of free vs. bound variables, just like in quantified expressions, integrals, etc.