Summation Formulas. Math Review. Let N > 0, let A, B, and C be constants, and let f and g be any functions. Then: S1: factor out constant

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1 Computer Science Dept Va Tech August McQuain WD Summation Formulas Let > 0, let A, B, and C e constants, and let f and g e any functions. Then: f C Cf ) ) S: factor out constant g f g f ) ) )) ) S: separate summed terms C C S3: sum of constant ) S4: sum of 6 ) ) S5: sum of squared 0 S6: sum of ^ ) S7: sum of ^-)

2 Logarithms Let e a real numer, > 0 and. Then, for any real numer > 0, the arithm of to ase is the power to which must e raised to yield. That is: ) y if and only if y For eample: 64) 6 ecause 6 64 /8) 3 ecause 3 /8 ) 0 ecause 0 If the ase is omitted, the standard convention in mathematics is that ase 0 is intended; in computer science the standard convention is that ase is intended. Computer Science Dept Va Tech August McQuain WD

3 Logarithms 3 Let a and e real numers, oth positive and neither equal to. Let > 0 and y > 0 e real numers. L: ) 0 L7: y) ) y) L: L3: L4: ) ) ) 0 for all 0 for all 0 L8: L9: y y ) y ) ) y) L5: L6: ) y ) y L0: ) a a ) ) Computer Science Dept Va Tech August McQuain WD

4 Limit of a Function Definition: 4 Let f) e a function with domain a, ) and let a < c <. The it of f) as approaches c is L if, for every positive real numer e, there is a positive real numer d such that whenever -c < d then f) L < e. The definition eing cumersome, the following theorems on its are useful. We assume f) is a function with domain as descried aove and that K is a constant. C: c K K c C: r r C3: c for all r 0 c c Computer Science Dept Va Tech August McQuain WD

5 Limit of a Function 5 Here assume f) and g) are functions with domain as descried aove and that K is a constant, and that oth the following its eist and are finite): c f ) A c g ) B Then: C4: c Kf ) K c f ) C5: C6: C7: c c c f ) g ) f ) g ) c c f )* g ) f )* g ) c c f ) / g ) f ) / g ) provided B 0 c c Computer Science Dept Va Tech August McQuain WD

6 Limit as Approaches Infinity Definition: 6 Let f) e a function with domain [0, ). The it of f) as approaches is L, where L is finite, if, for every positive real numer e, there is a positive real numer such that whenever > then f) L < e. The definition eing cumersome, the following theorems on its are useful. We assume f) is a function with domain [0, ) and that K is a constant. C8: K K 0 C9: C0: 0 for all r 0 r Computer Science Dept Va Tech August McQuain WD

7 Limit of a Rational Function 7 Given a rational function the last two rules are sufficient if a little algera is employed: Divide y highest power of from the denominator. Tae its term y term. Apply theorem C3. Computer Science Dept Va Tech August McQuain WD

8 Infinite Limits 8 In some cases, the it may e infinite. Mathematically, this means that the it does not eist. C: r for all r 0 C3: e C: Eample: Computer Science Dept Va Tech August McQuain WD

9 l'hôpital's Rule 9 In some cases, the reduction tric shown for rational functions does not apply: 7 5 ) 0 5?? In such cases, l'hôpital's Rule is often useful. If f) and g) are differentiale functions such that then: c f ) g ) c c f ) f ) g ) g ) c This also applies if the it is 0. Computer Science Dept Va Tech August McQuain WD

10 l'hôpital's Rule Eamples Applying l'hôpital's Rule: ) Another eample: 3 0 e 3 e 6 e 6 e 0 Recall that: f ) f ) D e e D f ) Computer Science Dept Va Tech August McQuain WD

11 Mathematical Induction Mathematical induction is a technique for proving that a statement is true for all integers in the range from 0 to, where 0 is typically 0 or. First or Wea) Principle of Mathematical Induction Let P) e a proposition regarding the integer, and let S e the set of all integers for which P) is true. If ) 0 is in S, and ) whenever is in S then + is also in S, then S contains all integers in the range [ 0, ). To apply the PMI, we must first estalish that a specific integer, 0, is in S estalishing the asis) and then we must estalish that if a aritrary integer, 0, is in S then its successor, +, is also in S. Computer Science Dept Va Tech August McQuain WD

12 Induction Eample Theorem: For all integers n, n +n is a multiple of. proof: Let S e the set of all integers for which n +n is a multiple of. If n =, then n +n =, which is oviously a multiple of. This estalishes the asis, that is in S. ow suppose that some integer is an element of S. Then + is a multiple of. We need to show that + is an element of S; in other words, we must show that +) ++) is a multiple of. Performing simple algera: +) ++) = + + ) + + ) = ow we now + is a multiple of, and the epression aove can e grouped to show: +) ++) = + ) + + ) = + ) + + ) The last epression is the sum of two multiples of, so it's also a multiple of. Therefore, + is an element of S. Therefore, y PMI, S contains all integers [, ). QED Computer Science Dept Va Tech August McQuain WD

13 Inadequacy of the First Form of Induction 3 Theorem: Every integer greater than 3 can e written as a sum of 's and 5's. That is, if > 3, then there are nonnegative integers and y such that = + 5y.) This is not easily) provale using the First Principle of Induction. The prolem is that the way to write + in terms of 's and 5's has little to do with the way is written in terms of 's and 5's. For eample, if we now that we can say that = + 5y + = + 5y + = + 5y ) = + 3) + 5y ) ut we have no reason to elieve that y is nonnegative. Suppose for eample that is 9.) Computer Science Dept Va Tech August McQuain WD

14 "Strong" Form of Induction 4 There is a second statement of induction, sometimes called the "strong" form, that is adequate to prove the result on the preceding slide: Second or Strong) Principle of Mathematical Induction Let P) e a proposition regarding the integer, and let S e the set of all integers for which P) is true. If ) 0 is in S, and ) whenever 0 through are in S then + is also in S, then S contains all integers in the range [ 0, ). Interestingly, the "strong" form of induction is ically equivalent to the "wea" form stated earlier; so in principle, anything that can e proved using the "strong" form can also e proved using the "wea" form. Computer Science Dept Va Tech August McQuain WD

15 Using the Second Form of Induction 5 Theorem: Every integer greater than 3 can e written as a sum of 's and 5's. proof: Let S e the set of all integers n > 3 for which n = + 5y for some nonnegative integers and y. If n = 4, then n = * + 5*0. If n = 5, then n = *0 + 5*. This estalishes the asis, that 4 and 5 are in S. ow suppose that all integers from 4 through are elements of S, where 5. We need to show that + is an element of S; in other words, we must show that + = r + 5s for some nonnegative integers r and s. ow + 6, so - 4. Therefore y our assumption, - = + 5y for some nonnegative integers and y. Then, simple algera yields that: + = - + = + 5y + = +) + 5y, whence + is an element of S. Therefore, y the Second PMI, S contains all integers [4, ). QED Computer Science Dept Va Tech August McQuain WD

16 Induction Eample Consider the sequences {a } and {d } : 6 a for 0 Then for all 0. a = d. proof: Let S = { 0 a = d }. d 3d d 0 0, d d for Trivial calculations show that a 0 = d 0 and a = d, so 0 and are in S. Suppose that there is some such that 0,,, are in S. In other words, assume that there is some such that a i = d i for all i from 0 to. ow, +, so from the definition of {d }and the inductive assumption we have: d 3d d 3 3 ) 3 ) Therefore + is in S, and so y the principle of induction, S = { 0}. QED Computer Science Dept Va Tech August McQuain WD