Summation Formulas Let > 0, let A, B, and C e constants, and let f and g e any functions. Then: k Cf ( k) C k S: factor out constant f ( k) k ( f ( k) ± g( k)) k S: separate summed terms f ( k) ± k g( k) k C C S3: sum of constant k ( + ) k S4: sum of k k k ( + )( 6 + S5: sum of k squared ) k 0 k + S6: sum of ^k k k k ( ) + S7: sum of k^(k-) Computer Science Dept Va Tech August 005 005 McQuain WD Logarithms Let e a real numer, > 0 and. Then, for any real numer > 0, the logarithm of to ase is the power to which must e raised to yield. That is: y log ( ) y if and only if For eample: log (64) 6 ecause log (/ 8) 3 ecause 6 3 64 / 8 log () 0 ecause 0 If the ase is omitted, the standard convention in mathematics is that log ase 0 is intended; in computer science the standard convention is that log ase is intended. Computer Science Dept Va Tech August 005 005 McQuain WD
Logarithms 3 Let a and e real numers, oth positive and neither equal to. Let > 0 and y > 0 e real numers. L: log () 0 L7: log ( y) log( ) + log( y) L: L3: L4: log ( ) log ( ) < 0 for all 0 < < log ( ) > 0 for all > L8: L9: log log( ) log( y) y y log ( ) y log ( ) L5: L6: log y ( ) log ( ) y L0: log ( ) loga ( ) log ( ) a Computer Science Dept Va Tech August 005 005 McQuain WD Limit of a Function 4 Definition: 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 ε, there is a positive real numer δ such that whenever -c < δ then f() L < ε. 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: K c K c C: r r C3: c for all r > 0 c c Computer Science Dept Va Tech August 005 005 McQuain WD
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 g( ) B c Then: C4: Kf ( ) K c 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 005 005 McQuain WD Limit as Approaches Infinity 6 Definition: 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 ε, there is a positive real numer such that whenever > then f() L < ε. 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 005 005 McQuain WD
Limit of a Rational Function 7 Given a rational function the last two rules are sufficient if a little algera is employed: 7 3 5 0 7 + + + 5 + 0 + + 5 5 3 + + 5 0 7 + + 5 3 + + 7 + 0 + 0 3 + 0 + 0 7 3 Divide y highest power of from the denominator. Take its term y term. Apply theorem C3. Computer Science Dept Va Tech August 005 005 McQuain WD 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: ( ) log Eample: 0 7 + 5 + 7 + 5 + 0 + 5 5 + 0 7 + 5 + 5 + Computer Science Dept Va Tech August 005 005 McQuain WD
l'hôpital's Rule 9 In some cases, the reduction trick shown for rational functions does not apply: 7 + 5log( ) + 0?? + 5 In such cases, l'hôpital's Rule is often useful. If f() and g() are differentiale functions such that then: f ( ) f ( ) c g( ) c g ( ) f ( ) g( ) c c This also applies if the it is 0. Computer Science Dept Va Tech August 005 005 McQuain WD l'hôpital's Rule Eamples Applying l'hôpital's Rule: 5 7 + 7 + 5log( ) + 0 + 5 7 0 Another eample: 3 + 0 3 e e 6 6 e e 0 Recall that: f ( ) f ( ) D[ e ] e D[ f ( ) ] Computer Science Dept Va Tech August 005 005 McQuain WD
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 Weak) Principle of Mathematical Induction Let P() e a proposition regarding the integer, and let S e the set of all integers k for which P(k) 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 005 005 McQuain WD 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 k is an element of S. Then k +k is a multiple of. We need to show that k+ is an element of S; in other words, we must show that (k+) +(k+) is a multiple of. Performing simple algera: (k+) +(k+) (k + k + ) + (k + ) k + 3k + ow we know k +k is a multiple of, and the epression aove can e grouped to show: (k+) +(k+) (k + k) + (k + ) (k + k) + (k + ) The last epression is the sum of two multiples of, so it's also a multiple of. Therefore, k+ is an element of S. Therefore, y PMI, S contains all integers [, ). QED Computer Science Dept Va Tech August 005 005 McQuain WD
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 know that + 5y we can say that + + 5y + + 5(y ) + 5 + ( + 3) + 5(y ) ut we have no reason to elieve that y is nonnegative. (Suppose for eample that is 9.) Computer Science Dept Va Tech August 005 005 McQuain WD "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 k for which P(k) 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 logically equivalent to the "weak" form stated earlier; so in principle, anything that can e proved using the "strong" form can also e proved using the "weak" form. Computer Science Dept Va Tech August 005 005 McQuain WD
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 k are elements of S, where k 5. We need to show that k+ is an element of S; in other words, we must show that k+ r + 5s for some nonnegative integers r and s. ow k+ 6, so k- 4. Therefore y our assumption, k- + 5y for some nonnegative integers and y. Then, simple algera yields that: k+ k- + + 5y + (+) + 5y, whence k+ is an element of S. Therefore, y the Second PMI, S contains all integers [4, ). QED Computer Science Dept Va Tech August 005 005 McQuain WD Induction Eample Consider the sequences {a k } and {d k } : k a for k 0 k Then for all k 0. a k d k. d0 0, d dk 3d k dk for k 6 proof: Let S {k 0 a k d k }. 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 k }and the inductive assumption we have: d 3d d + 3( 3 ) ( 3 + ) + Therefore + is in S, and so y the principle of induction, S {k 0}. QED Computer Science Dept Va Tech August 005 005 McQuain WD