Concrete mathematics : a foundation for computer science

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60 SUMS In fact, our definition of infinite sums is quite simple. Let K be any set, and let ok be a real-valued term defined for each k E K. (Here ‘k’ might actually stand for several indices kl , k2, . . , and K might therefore be multidimensional.) Any real number x can be written as the difference of its positive and negative parts, x .= x+-x where x+ =x.[x>O] and x- = -x.[x 0 and each term ok is nonnegative. The proof
2.7 INFINITE SUMS 61 in this special case works because tkEF cok = c tkeF ok for all finite Sets F; the latter fact follows by induction on the size of F. The associative law (2.16) can be stated as follows: If tkEK ok and tkeK bk converge absolutely to A and B, respectively, then tkek(ok + bk) converges absolutely to A + B. This turns out to be a special case of a more general theorem that we will prove shortly. The commutative law (2.17) doesn’t really need to be proved, because we have shown in the discussion following (2.35) how to derive it as a special case of a general rule for interchanging the order of summation. The main result we need to prove is the fundamental principle of multiple sums: Absolutely convergent sums over two or more indices can always be summed first with respect to any one of those indices. Formally, we shall Best to skim this prove that if J and the elements of {Ki 1 j E J} are any sets of indices such that page the first time you get here. - Your friendly TA x iEJ kEKj oi,k converges absolutely to A, then there exist complex numbers Aj for each j E J such that IL &K, oj,k converges absolutely to Aj, and t Aj converges absolutely to A. iEJ It suffices to prove this assertion when all terms are nonnegative, because we can prove the general case by breaking everything into real and imaginary, positive and negative parts as before. Let’s assume therefore that oi,k 3 0 for all pairs (j, k) E M, where M is the master index set {(j, k) 1 j E J, k E Kj}. We are given that tCj,k)EM oj,k is finite, namely that L aj,k 6 A (j.k)EF for all finite subsets F C M, and that A is the least such upper bound. If j is any element of J, each sum of the form xkEFi oj,k where Fj is a finite subset of Kj is bounded above by A. Hence these finite sums have a least upper bound Ai 3 0, and tkEKi oj,k = Aj by definition. We still need to prove that A is the least upper bound of xjEG Aj, for all finite subsets G G J. Suppose that G is a finite subset of J with xjEG Aj = A’ > A. We CXI find finite subsets Fi c Kj such that tkeFi oj,k > (A/A’)Aj for each j E G with Aj > 0. There is at least one such j. But then ~.iEG,kCFi oj,k > (A/A’) xjEG Aj = A, contradicting the fact that we have
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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
Page 24 and 25: 10 RECURRENT PROBLEMS But what abou
Page 26 and 27: 12 RECURRENT PROBLEMS that is, in t
Page 28 and 29: 14 RECURRENT PROBLEMS by separating
Page 30 and 31: 16 RECURRENT PROBLEMS For example,
Page 32 and 33: 18 RECURRENT PROBLEMS Homework exer
Page 34 and 35: 20 RECURRENT PROBLEMS 21 Suppose th
Page 36 and 37: 22 SUMS The three-dots notation has
Page 38 and 39: 24 SUMS A slightly modified form of
Page 40 and 41: 26 SUMS where A(n), B(n), and C(n)
Page 42 and 43: 28 SUMS Let’s apply these ideas t
Page 44 and 45: 30 SUMS 2.3 MANIPULATION OF SUMS No
Page 46 and 47: 32 SUMS Typically we use rule (2.20
Page 48 and 49: 34 SUMS because the derivative of a
Page 50 and 51: 36 SUMS instead of summing over all
Page 52 and 53: 38 SUMS The second sum here is zero
Page 54 and 55: 40 SUMS The normal way to evaluate
Page 56 and 57: 42 SUMS First, as usual, we look at
Page 58 and 59: 44 SUMS order to get an equation fo
Page 60 and 61: 46 SUMS One way to use this fact is
Page 62 and 63: 48 SUMS definition where the factor
Page 64 and 65: 50 SUMS Let’s try to recap this;
Page 66 and 67: 52 SUMS we need an appropriate defi
Page 68 and 69: 54 SUMS This formula indicates why
Page 70 and 71: 56 SUMS u(x) = x and Av(x) = 2’;
Page 72 and 73: 58 SUMS The definition in the previ
Page 76 and 77: 62 SUMS tCj,kiEF a. J, k < A for al
Page 78 and 79: 64 SUMS 18 Let 9%~ and Jz be the re
Page 80 and 81: 66 SUMS 34 35 36 Prove that if the
Page 82 and 83: 68 INTEGER FUNCTIONS below the line
Page 84 and 85: 70 INTEGER FUNCTIONS same inequalit
Page 86 and 87: 72 INTEGER FUNCTIONS An important s
Page 88 and 89: 74 INTEGER FUNCTIONS By the way, th
Page 90 and 91: 76 INTEGER FUNCTIONS (Previously K
Page 92 and 93: 78 INTEGER FUNCTIONS This derivatio
Page 94 and 95: 80 INTEGER FUNCTIONS We’ve got mo
Page 96 and 97: 82 INTEGER FUNCTIONS division, and
Page 98 and 99: 84 INTEGER FUNCTIONS more than one
Page 100 and 101: 86 INTEGER FUNCTIONS 3.5 FLOOR/CEIL
Page 102 and 103: 88 INTEGER FUNCTIONS For this purpo
Page 104 and 105: 90 INTEGER FUNCTIONS it comes out i
Page 106 and 107: 92 INTEGER FUNCTIONS To summarize,
Page 108 and 109: 94 INTEGER FUNCTIONS Also, as we gu
Page 110 and 111: 96 INTEGER FUNCTIONS Basics 10 Show
Page 112 and 113: 98 INTEGER FUNCTIONS 31 Prove or di
Page 114 and 115: 100 INTEGER FUNCTIONS 43 Find an in
Page 116 and 117: 4 Number Theory INTEGERS ARE CENTRA
Page 118 and 119: 104 NUMBER THEORY The left side can
Page 120 and 121: 106 NUMBER THEORY product of primes
Page 122 and 123: 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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222 BINOMIAL COEFFICIENTS One use o
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224 BINOMIAL COEFFICIENTS Therefore
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226 BINOMIAL COEFFICIENTS Now f(-b)
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228 BINOMIAL COEFFICIENTS negative
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230 BINOMIAL COEFFICIENTS valid for
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232 BINOMIAL COEFFICIENTS 18 Find a
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234 BINOMIAL COEFFICIENTS 37 Show t
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236 BINOMIAL COEFFICIENTS 57 Use Go
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238 BINOMIAL COEFFICIENTS 72 Prove
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240 BINOMIAL COEFFICIENTS 86 Let al
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242 BINOMIAL COEFFICIENTS Research
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244 SPECIAL NUMBERS Table 244 Stirl
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246 SPECIAL NUMBERS There are eleve
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248 SPECIAL NUMBERS cycle arrangeme
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250 SPECIAL NUMBERS Table 250 Basic
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252 SPECIAL NUMBERS We can remember
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254 SPECIAL NUMBERS Table 254 Euler
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{'l 'I L\‘, 0 1 / 2 256 SPECIAL N
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258 SPECIAL NUMBERS Table 258 Stirl
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260 SPECIAL NUMBERS ’ ’ ’ (Th
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262 SPECIAL NUMBERS we found that a
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264 SPECIAL NUMBERS Rearranging, we
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266 SPECIAL NUMBERS Thus we have th
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268 SPECIAL NUMBERS (We have used t
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270 SPECIAL NUMBERS Bernoulli numbe
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272 SPECIAL NUMBERS trigonometric f
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274 SPECIAL NUMBERS Recurrence (6.9
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276 SPECIAL NUMBERS Gnethingwecando
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278 SPECIAL NUMBERS cases are: a0 =
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280 SPECIAL NUMBERS The process of
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282 SPECIAL NUMBERS Fk < n < Fk+l;
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284 SPECIAL NUMBERS We have now boi
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286 SPECIAL NUMBERS Before we stop
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288 SPECIAL NUMBERS The continuant
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290 SPECIAL NUMBERS in four steps:
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292 SPECIAL NUMBERS Thus we can con
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294 SPECIAL NUMBERS numbers are 1,
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296 SPECIAL NUMBERS 6 An explorer h
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298 SPECIAL NUMBERS 24 Prove that t
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300 SPECIAL NUMBERS 44 Prove the co
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302 SPECIAL NUMBERS 61 Prove the id
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304 SPECIAL NUMBERS 80 Show that co
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7 Generating Functions THE MOST POW
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308 GENERATING FUNCTIONS are added
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310 GENERATING FUNCTIONS (The last
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312 GENERATING FUNCTIONS Now we hav
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314 GENERATING FUNCTIONS The first
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316 GENERATING FUNCTIONS determine
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318 GENERATING FUNCTIONS operation
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320 GENERATING FUNCTIONS Table 320
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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,
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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
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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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