CSCI2110 Tutorial 6: More Counting by Mapping, Number Sequences

1. CSCI2110 Tutorial 6:More Counting by Mapping, Number Sequences Chow Chi Wang (cwchow ‘at’ cse.cuhk.edu.hk)

2. More Counting by Mapping

3. Division Rule • If a function f from A to B is k-to-1(this means for every element in Bis mapped by exactly k elements in A.), then . • Sometimes we can’t find a direct way to count the size of a set A. • The idea of the division rule is to establish a k-to-1 correspondence between A and another set B, which is hopefully more easily countable.

4. Division Rule (Example) • (1) Consider the string NILLAPALOO in this question. • How many distinguishable ways can the letters be arranged in order? • Let A be the set of all possible rearrangement of the string. • Let B be the set of the strings of length 10 constructed by these symbols S = {N, I, L1, L2, L3, A1,A2, P, O1, O2}. • Consider the function , where f replaces every symbols in S by the corresponding letter. • e.g. f(NIL1L2L3A1A2PO1O2) = NILLLAAPOO

5. Division Rule (Example) • We know that: • Because there are 10 choices for the 1st component of the string • There are 9 choices for the 2nd component of the string • … • So in total there are 10x9x…x1 = 10! many possible strings in B. • f is a -to-1function. • Strings of the forms and map to the same string in A by f. And there are 2! such forms corresponds to ’s. • Similarly for the ’s and ’s, which have 3! and 2! different forms to consider respectively. • So by the division rule, we know that:

6. Number Sequences

7. Sum of a Sequence (Series) • (Sigma Notation) . • Some common facts: • . • . • .

8. Telescoping Series • A telescoping series is a series whose partial sums eventually only have a fixed number of terms after cancellation. • e.g.

9. Telescoping Series (Puzzle) • 0 = 1? • Consider: ! • What is wrong?

10. Arithmetic Sequence • An arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant • e.g. 1, 3, 5, 7, 9, … • In general, arithmetic sequence can be expressed as: • (initial value) • for ( is called the common difference)

11. Arithmetic Series • Let and is an arithmetic sequence. • We have: • Because is an arithmetic sequence, we have:

12. Arithmetic Series • Because there are such terms so: Therefore:

13. Geometric Sequence • A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio. • e.g. 1, 2, 4, 8, 16, … • In general, arithmetic sequence can be expressed as: • (initial value) • for ( is called the common ratio)

14. Geometric Series • Let and is a geometric sequence. • We have: • Therefore, for :

15. Harmonic Series • We define to be the n-th harmonic number. • Harmonic series is divergent! This means • This is because we have: • So we have , therefore harmonic series is divergent.

16. Harmonic Series • Here is a bound for the harmonic numbers: • We can prove this by integration.

17. Pi Notation • (Pi Notation) .

18. Thank You!