Concrete mathematics : a foundation for computer science

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272 SPECIAL NUMBERS trigonometric functions in terms of their hyperbolic cousins by using the rules sin z = -isinh iz , cos z = cash iz; the corresponding power series are sin2 = 2’ 1!-3!+5!--... 23 25 2’ 23 25 , sinhz = T+“j-i.+5r+...; 20 22 24 cosz = o!-2!+4?--...) .ci .; zi coshz = ol+2r+T+... . . . . Hence cot z = cos z/sin z = i cash iz/ sinh iz = i coth iz, and we have Another remarkable formula for zcot z was found by Euler (exercise 73): zcotz = l-2tTg. k>,krr -z2 We can expand Euler’s formula in powers of z2, obtaining . (6.86) (6.87) (6.88) Equating coefficients of zZn with those in our other formula, (6.87), gives us an almost miraculous closed form for infinitely many infinite sums:
6.5 BERNOULLI NUMBERS 273 And that’s not all. Bernoulli numbers also appear in the coefficients of the tangent function, (6.92) as well as other trigonometric functions (exercise 70). Formula (6.92) leads to another important fact about the Bernoulli numbers, namely that T2n-, = (-1)-l 4n(4n-l) 2n We have, for example: n 135 7 9 11 13 Tll 1 2 16 272 7936 353792 22368256 Bzn is a positive integer. (Wi) (The T's are called tangent numbers.) One way to prove (6.g3), following an idea of B. F. Logan, is to consider the power series sinz+xcosz - x+ (l+x2)z+ (2x3+2x); + (6x4+8x2+2); + cosz-xsinz - When x = tanw, where T,,(x) is a polynomial in x; setting x = 0 gives T, (0) = Tn, the nth this is tan( z + w) . tangent number. If we differentiate (6.94) with respect to x, we get 1 (cosz-xsinz)2 = xT(x)$; Tl>O but if we differentiate with respect to z, we get 1+x2 (cosz-xsin~)~ = tT,(xl& = tT,_M$. ll>l tl)O (Try it-the cancellation is very pretty.) Therefore we have -&,+1(x) = (1 +x2)T;(x), To(x) = x, (fhd a simple recurrence from which it follows that the coefficients of Tn(x) are nonnegative integers. Moreover, we can easily prove that Tn(x) has degree n + 1, and that its coefficients are alternately zero and positive. Therefore Tz,+I (0) = Tin+, is a positive integer, as claimed in (6.93).
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CONCRETE MATHEMATICS Dedicated to L
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CONCRETE MATHEMATICS Ronald L. Grah
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“A odience, level, and treatment
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‘I a concrete life preserver thro
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I’m unaccustomed to this face. De
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n [ n-l 1 n {I m Prestressed concre
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CONTENTS xiii 5.3 Tricks of the Tra
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2 RECURRENT PROBLEMS It’s not imm
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4 RECURRENT PROBLEMS Of course the
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6 RECURRENT PROBLEMS through any of
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8 RECURRENT PROBLEMS From these sma
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10 RECURRENT PROBLEMS But what abou
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12 RECURRENT PROBLEMS that is, in t
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14 RECURRENT PROBLEMS by separating
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16 RECURRENT PROBLEMS For example,
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18 RECURRENT PROBLEMS Homework exer
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20 RECURRENT PROBLEMS 21 Suppose th
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22 SUMS The three-dots notation has
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24 SUMS A slightly modified form of
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26 SUMS where A(n), B(n), and C(n)
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28 SUMS Let’s apply these ideas t
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30 SUMS 2.3 MANIPULATION OF SUMS No
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32 SUMS Typically we use rule (2.20
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34 SUMS because the derivative of a
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36 SUMS instead of summing over all
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38 SUMS The second sum here is zero
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40 SUMS The normal way to evaluate
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42 SUMS First, as usual, we look at
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44 SUMS order to get an equation fo
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46 SUMS One way to use this fact is
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48 SUMS definition where the factor
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50 SUMS Let’s try to recap this;
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52 SUMS we need an appropriate defi
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54 SUMS This formula indicates why
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56 SUMS u(x) = x and Av(x) = 2’;
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58 SUMS The definition in the previ
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60 SUMS In fact, our definition of
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62 SUMS tCj,kiEF a. J, k < A for al
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64 SUMS 18 Let 9%~ and Jz be the re
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66 SUMS 34 35 36 Prove that if the
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68 INTEGER FUNCTIONS below the line
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70 INTEGER FUNCTIONS same inequalit
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72 INTEGER FUNCTIONS An important s
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74 INTEGER FUNCTIONS By the way, th
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76 INTEGER FUNCTIONS (Previously K
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78 INTEGER FUNCTIONS This derivatio
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80 INTEGER FUNCTIONS We’ve got mo
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82 INTEGER FUNCTIONS division, and
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84 INTEGER FUNCTIONS more than one
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86 INTEGER FUNCTIONS 3.5 FLOOR/CEIL
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88 INTEGER FUNCTIONS For this purpo
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90 INTEGER FUNCTIONS it comes out i
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92 INTEGER FUNCTIONS To summarize,
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94 INTEGER FUNCTIONS Also, as we gu
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96 INTEGER FUNCTIONS Basics 10 Show
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98 INTEGER FUNCTIONS 31 Prove or di
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100 INTEGER FUNCTIONS 43 Find an in
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4 Number Theory INTEGERS ARE CENTRA
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104 NUMBER THEORY The left side can
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106 NUMBER THEORY product of primes
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108 NUMBER THEORY only finitely man
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110 NUMBER THEORY But 2P - 1 isn’
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112 NUMBER THEORY arrangements in a
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114 NUMBER THEORY The binary repres
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116 NUMBER THEORY A fraction m/n is
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118 NUMBER THEORY A mediant fractio
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120 NUMBER THEORY This representati
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122 NUMBER THEORY This means that w
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124 NUMBER THEORY Since x mod m dif
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126 NUMBER THEORY than to say that
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128 NUMBER THEORY require division,
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130 NUMBER THEORY Now we must show
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132 NUMBER THEORY And it’s possib
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134 NUMBER THEORY For example, (~(1
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136 NUMBER THEORY Conversely, if f(
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138 NUMBER THEORY (We must add 1 to
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140 NUMBER THEORY The problem of co
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142 NUMBER THEORY n is odd, n4 and
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144 NUMBER THEORY But the inner sum
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146 NUMBER THEORY 22 The number 111
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148 NUMBER THEORY 40 If the radix p
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150 NUMBER THEORY 57 Let S(m,n) be
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152 NUMBER THEORY 73 If the 0(n) +
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154 BINOMIAL COEFFICIENTS For examp
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156 BINOMIAL COEFFICIENTS And now t
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158 BINOMIAL COEFFICIENTS property
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160 BINOMIAL COEFFICIENTS (3, then
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162 BINOMIAL COEFFICIENTS Binomial
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164 BINOMIAL COEFFICIENTS Table 164
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166 BINOMIAL COEFFICIENTS not rows.
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168 BINOMIAL COEFFICIENTS Equation
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170 BINOMIAL COEFFICIENTS that m =
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172 BINOMIAL COEFFICIENTS ifweuse (
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176 BINOMIAL COEFFICIENTS sum on th
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178 BINOMIAL COEFFICIENTS some data
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180 BINOMIAL COEFFICIENTS Problem 4
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182 BINOMIAL COEFFICIENTS We’re l
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184 BINOMIAL COEFFICIENTS We can’
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186 BINOMIAL COEFFICIENTS 5.3 TRICK
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188 BINOMIAL COEFFICIENTS if we app
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190 BINOMIAL COEFFICIENTS The Newto
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192 BINOMIAL COEFFICIENTS One examp
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194 BINOMIAL COEFFICIENTS We can de
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196 BINOMIAL COEFFICIENTS This obse
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198 BINOMIAL COEFFICIENTS Generatin
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200 BINOMIAL COEFFICIENTS can be pu
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204 BINOMIAL COEFFICIENTS This hold
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206 BINOMIAL COEFFICIENTS It’s ou
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208 BINOMIAL COEFFICIENTS into this
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210 BINOMIAL COEFFICIENTS integer.
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212 BINOMIAL COEFFICIENTS into to b
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214 BINOMIAL COEFFICIENTS Now sin(
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216 BINOMIAL COEFFICIENTS But look
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218 BINOMIAL COEFFICIENTS It’s al
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220 BINOMIAL COEFFICIENTS A similar
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Page 238 and 239: 224 BINOMIAL COEFFICIENTS Therefore
Page 240 and 241: 226 BINOMIAL COEFFICIENTS Now f(-b)
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Page 244 and 245: 230 BINOMIAL COEFFICIENTS valid for
Page 246 and 247: 232 BINOMIAL COEFFICIENTS 18 Find a
Page 248 and 249: 234 BINOMIAL COEFFICIENTS 37 Show t
Page 250 and 251: 236 BINOMIAL COEFFICIENTS 57 Use Go
Page 252 and 253: 238 BINOMIAL COEFFICIENTS 72 Prove
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Page 256 and 257: 242 BINOMIAL COEFFICIENTS Research
Page 258 and 259: 244 SPECIAL NUMBERS Table 244 Stirl
Page 260 and 261: 246 SPECIAL NUMBERS There are eleve
Page 262 and 263: 248 SPECIAL NUMBERS cycle arrangeme
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Page 266 and 267: 252 SPECIAL NUMBERS We can remember
Page 268 and 269: 254 SPECIAL NUMBERS Table 254 Euler
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Page 274 and 275: 260 SPECIAL NUMBERS ’ ’ ’ (Th
Page 276 and 277: 262 SPECIAL NUMBERS we found that a
Page 278 and 279: 264 SPECIAL NUMBERS Rearranging, we
Page 280 and 281: 266 SPECIAL NUMBERS Thus we have th
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Page 284 and 285: 270 SPECIAL NUMBERS Bernoulli numbe
Page 288 and 289: 274 SPECIAL NUMBERS Recurrence (6.9
Page 290 and 291: 276 SPECIAL NUMBERS Gnethingwecando
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Page 294 and 295: 280 SPECIAL NUMBERS The process of
Page 296 and 297: 282 SPECIAL NUMBERS Fk < n < Fk+l;
Page 298 and 299: 284 SPECIAL NUMBERS We have now boi
Page 300 and 301: 286 SPECIAL NUMBERS Before we stop
Page 302 and 303: 288 SPECIAL NUMBERS The continuant
Page 304 and 305: 290 SPECIAL NUMBERS in four steps:
Page 306 and 307: 292 SPECIAL NUMBERS Thus we can con
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Page 310 and 311: 296 SPECIAL NUMBERS 6 An explorer h
Page 312 and 313: 298 SPECIAL NUMBERS 24 Prove that t
Page 314 and 315: 300 SPECIAL NUMBERS 44 Prove the co
Page 316 and 317: 302 SPECIAL NUMBERS 61 Prove the id
Page 318 and 319: 304 SPECIAL NUMBERS 80 Show that co
Page 320 and 321: 7 Generating Functions THE MOST POW
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Page 324 and 325: 310 GENERATING FUNCTIONS (The last
Page 326 and 327: 312 GENERATING FUNCTIONS Now we hav
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Page 332 and 333: 318 GENERATING FUNCTIONS operation
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322 GENERATING FUNCTIONS middle of
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324 GENERATING FUNCTIONS Fortunatel
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326 GENERATING FUNCTIONS Once we’
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328 GENERATING FUNCTIONS Step 1 is
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330 GENERATING FUNCTIONS nice prope
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332 GENERATING FUNCTIONS Finally, s
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334 GENERATING FUNCTIONS Step 3 was
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336 GENERATING FUNCTIONS This is a
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338 GENERATING FUNCTIONS Table 337
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340 GENERATING FUNCTIONS F(z)' = i
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342 GENERATING FUNCTIONS We can app
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344 GENERATING FUNCTIONS Step 3 is
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346 GENERATING FUNCTIONS Raney’s
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348 GENERATING FUNCTIONS Notice tha
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350 GENERATING FUNCTIONS Chapter 5
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352 GENERATING FUNCTIONS Now let’
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354 GENERATING FUNCTIONS function f
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356 GENERATING FUNCTIONS 7.7 DIRICH
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358 GENERATING FUNCTIONS 5 Find a g
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360 GENERATING FUNCTIONS 20 A power
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362 GENERATING FUNCTIONS 29 What is
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364 GENERATING FUNCTIONS 43 The New
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366 GENERATING FUNCTIONS 53 The seq
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368 DISCRETE PROBABILITY total of 6
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370 DISCRETE PROBABILITY about R if
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372 DISCRETE PROBABILITY The mean o
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374 DISCRETE PROBABILITY more sprea
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376 DISCRETE PROBABILITY hence VS =
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378 DISCRETE PROBABILITY And we can
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380 DISCRETE PROBABILITY variance b
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382 DISCRETE PROBABILITY (1 +z+...
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384 DISCRETE PROBABILITY ~1 and ~2
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386 DISCRETE PROBABILITY where Ug i
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388 DISCRETE PROBABILITY Repeating
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390 DISCRETE PROBABILITY There’s
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392 DISCRETE PROBABILITY But the co
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394 DISCRETE PROBABILITY In the spe
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396 DISCRETE PROBABILITY Here 1 is
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398 DISCRETE PROBABILITY For exampl
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400 DISCRETE PROBABILITY Thus the m
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402 DISCRETE PROBABILITY Let sj be
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404 DISCRETE PROBABILITY Therefore
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406 DISCRETE PROBABILITY The mean v
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408 DISCRETE PROBABILITY Complicate
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410 DISCRETE PROBABILITY all k and
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412 DISCRETE PROBABILITY to be a pg
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414 DISCRETE PROBABILITY 10 What’
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416 DISCRETE PROBABILITY 28 What is
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418 DISCRETE PROBABILITY 38 What is
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420 DISCRETE PROBABIL1T.Y a Once he
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422 DISCRETE PROBABILITY 50 Conside
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424 DISCRETE PROBABILITY 58 Are the
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426 ASYMPTOTICS The word asymptotic
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428 ASYMPTOTIC3 How about the funct
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430 ASYMPTOTICS N. G. de Bruijn beg
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432 ASYMPTOTICS subtle consideratio
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434 ASYMPTOTICS inefficient “beca
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436 ASYMPTOTICS 9.3 0 MANIPULATION
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138 ASYMPTOTICS Table 438 Asymptoti
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440 ASYMPTOTICS Notice how the 0 te
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442 ASYMPTOTICS Problem 3: The nth
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444 ASYMPTOTICS Problem 4: A sum fr
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446 ASYMPTOTICS Problem 5: An infin
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448 ASYMPTOTICS We might as well do
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450 ASYMPTOTIC3 a rough estimate an
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452 ASYMPTOTICS This last estimate
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454 ASYMPTOTICS Here’s another ex
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456 ASYMPTOTICS On the left is a ty
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458 ASYMPTOTICS (The Bernoulli poly
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460 ASYMPTOTICS The graph of B,(x)
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462 ASYMPTOTICS 9.6 FINAL SUM:MATIO
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464 ASYMPTOTICS The integral JT B4(
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466 ASYMPTOTIC3 where 0 < a,,,, < 1
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468 ASYMPTOTICS In Chapter 5, equat
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470 ASYMPTOTICS For example, we hav
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472 ASYMPTOTICS The final task seem
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474 ASYMPTOTICS This is our approxi
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476 ASYMPTOTIC23 9 Prove (9.22) rig
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478 ASYMPTOTICS 39 Evaluate xOsk
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480 ASYMPTOTICS a What is the asymp
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482 ASYMPTOTIC3 Research problems 6
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484 ANSWERS TO EXERCISES Venn [294]
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486 ANSWERS TO EXERCISES move the b
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488 ANSWERS TO EXERCISES 2.2 This i
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490 ANSWERS TO EXERCISES 2.24 Summi
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492 ANSWERS TO EXERCISES 2.36 (a) B
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494 ANSWERS TO EXERCISES 3.21 If 10
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496 ANSWERS TO EXERCISES the circle
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498 ANSWERS TO EXERCISES 3.40 Let L
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500 ANSWERS TO EXERCISES 3.50 H. S.
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502 ANSWERS TO EXERCISES 4.22 (b,
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504 ANSWERS TO EXERCISES 4.40 Let f
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506 ANSWERS TO EXERCISES 4.50 (a) I
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508 ANSWERS TO EXERCISES and (k- 1)
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510 ANSWERS TO EXERCISES In both ca
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512 ANSWERS TO EXERCISES 5.7 Yes, b
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514 ANSWERS TO EXERCISES because al
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516 ANSWERS TO EXERCISES 5.44 Cance
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518 ANSWERS TO EXERCISES and this i
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520 ANSWERS TO EXERCISES The infini
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522 ANSWERS TO EXERCISES 5.82 Let ~
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524 ANSWERS TO EXERCISES Since ~06j
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526 ANSWERS TO EXERCISJES 5.94 This
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528 ANSWERS TO EXERCISES 6.21 The h
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530 ANSWERS TO EXERCISIES 6.40 If 6
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532 ANSWERS TO EXERCISES solutions
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534 ANSWERS TO EXERCISES 6.62 (a) A
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536 ANSWERS TO EXERCISES 6.74 If p(
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538 ANSWERS TO EXERCISIES One of th
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540 ANSWERS TO EXERCISES 6.85 The p
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542 ANSWERS TO EXERCISES Hence gzn+
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544 ANSWERS TO EXERCISES 7.24 ntk,+
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546 ANSWERS TO EXERCISES 7.40 The e
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548 ANSWERS TO EXERCISES 1 + a. Hen
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550 ANSWERS TO EXERCISES applied to
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552 ANSWERS TO EXERCISES 8.13 (Solu
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554 ANSWERS TO EXERCISES 8.24 (a) A
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556 ANSWERS TO EXERCISES 8.32 By sy
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558 ANSWERS TO EXERCISES 8.37 The n
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560 ANSWERS TO EXERCISES probabilit
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562 ANSWERS TO EXERCISES And since
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564 ANSWERS TO EXERCISES 8.56 If m
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566 ANSWERS TO EXERCISES We can now
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568 ANSWERS TO EXERCISES 9.19 Hlo =
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570 ANSWERS TO EXERCIS:ES 9.30 Let
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572 ANSWERS TO EXERCISES 9.42 The h
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574 ANSWERS TO EXERCISIES length of
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576 ANSWERS TO EXERCISES 9.58 Let 0
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B Bibliography HERE ARE THE WORKS c
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580 BIBLIOGRAPHY 24' 25 26 27 28 29
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582 BIBLIOGRAPHY 51 52 53 54 55 56
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584 BIBLIOGRAPHY 78 79 80 81 82 83
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586 BIBLIOGRAPHY 100 R. A. Fisher,
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588 BIBLIOGRAPHY 128 Ronald L. Grah
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590 BIBLIOGRAPHY 160 K. Inkeri, “
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592 BIBLIOGRAPHY 188 E.E. Kummer,
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594 BIBLIOGRAPHY 214 Z.A. Melzak, C
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596 BIBLIOGRAPHY 243 George N. Rane
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598 BIBLIOGRAPHY 271 A. D. Solov’
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600 BIBLIOGRAPHY 299 Louis Weisner,
Page 616 and 617:
602 CREDITS FOR EXERCISES 1.1 1.2 1
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604 CREDITS FOR EXERCISES 6.46 6.47
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Index WHEN AN INDEX ENTRY refers to
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608 INDEX Bois-Reymond, Paul David
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610 INDEX Degenerate hypergeometric
Page 626 and 627:
612 INDEX Feder, Tom&s, 604. Feigen
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614 INDEX Hall, Marshall, Jr., 588.
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616 INDEX Lincoln, Abraham, 387. Li
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618 INDEX Patashnik, Oren, iii, iv,
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620 INDEX Running time, 411-412. Ru
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622 INDEX Sylvester, James Joseph,
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List of Tables Sums and differences
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