Discrrete mathematics for computer science series

Series Geometric Series Binomial Coefficients Harmonic Series Sum of a Geometric Series • What is   1      • Method 1: Prove by induction that for every n≥0, 1                 • And then make some argument about the limit as n →∞ to conclude that the sum is Sum of a Geometric Series        • Another way Recall that 1  and plug in x = ½ ! These “formal power series” have many uses Another example    • What isX             • Since       1           1          ⎛⎞  ⎜ ⎟ ⎝⎠ Another Example    • What is S            • Generalize Note that S=F(1/3) where F(x)           Manipulating Power Series Since F(x)                                                              ⎛⎞ ⎜ ⎟     ⎝ ⎠ ⎛  ⎞  ⎜ ⎝ ⎟ ⎠ Another Approach F(x)           • But then       1  d        dx 1                           Identities involving “Choose” ⎛ n ⎞ • What is  ⎜  ⎟  ⎝ i ⎠  • “Set Theory” derivation – Let S be a set of size n – This is the sum of the number of element subsets, plus the number of 1element subsets, plus …, plus the number of n-element subsets – Total 2n Binomial Theorem ⎛  ⎞    (x     ⎜   ⎟  ⎝  ⎠   because if you multiply out (x+y)(x+y)(x+y)…(x+y) the coefficient of xiyn-i is the number of different ways of choosing x from i factors and y from n-i factors Using the Binomial Theorem ⎛  ⎞      Since (x     ⎜ ⎟  ⎝  ⎠   substituting x=y=1 yields  ⎛ ⎞ ⎛  ⎞       (1      ⎜     ⎜ ⎟ ⎟   ⎠ ⎠  ⎝  ⎝  Using the Binomial Theorem What is ⎛  ⎞   ⎜⎝  ⎟⎠    i Try to make this look like the binomial thm   ⎛ ⎞ ⎛  ⎞     i               ⎜⎝  ⎟⎠  ⎜⎝  ⎟⎠   Stacking books Can you stack identical books so the top one is completely off the table? 3/22/19 Stacking books • Can you stack identical books so the top one is completely off the table? • One book: balance at middle 3/22/19 Stacking books • Two books: balance top one at the middle, the second over the table edge at the center of gravity of the pair 1 ½ 3/22/19 Stacking books  Three books: balance top one at the middle, the second over the third at ½, and the trio over the table edge at the center of gravity ½ 3/22/19 Stacking books  Three books: balance top one at the middle, the second over the third at ½, and the trio over the table edge at the center of gravity = (1+2+2½)/3 = 1⅚ from right end 1⅚ ½ 3/22/19 Stacking books  Three books: balance top one at the middle, the second over the third at ½, and the trio over the table edge at the center of gravity = (1+2+2½)/3 = 1⅚ from right end 1⅚ ⅓ ½ See a pattern? 3/22/19 The Harmonic Series Diverges   • Let H n   Then for any n,    H 2n     • Doubling the number of terms adds at least ẵ to the sum Corollary The series diverges, that is, For every m         3/22/19 Proof by WOP that the Harmonic Series Diverges  Let P(n)            Suppose C is nonempty. Let m be the minimal element of C                   3/22/19 The Harmonic Series Diverges But then H 2m 3/22/19                                                           So with a stack of 31 books you get book lengths off the table! And there is no limit to how far the stack can extend! http://mathforum.org/advanced/robertd/harmonic.html 3/22/19 FINIS ... Harmonic Series Diverges   • Let H n   Then for any n,    H 2n     • Doubling the number of terms adds at least ½ to the sum • Corollary The series diverges, that is, For every m... to conclude that the sum is Sum of a Geometric Series        • Another way Recall that 1  and plug in x = ½ ! These “formal power series have many uses Another example    • What...Sum of a Geometric Series • What is   1      • Method 1: Prove by induction that for every n≥0, 1                 • And then