Concrete mathematics : a foundation for computer science

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462 ASYMPTOTICS 9.6 FINAL SUM:MATIONS Now comes the summing up, as we prepare to conclude this book. We will apply Euler’s summation formula to some interesting and important examples. Summation 1: This one jis too easy. But first we will consider an interesting unimportant example, namely a sum that we already know how to do. Let’s see what Euler’s summation formula tells us if we apply i.t to the telescoping sum sn= EL=: l
9.6 FINAL SUMMATIONS 463 This is kind of a mess; it certainly doesn’t look like the real answer 1 - n-l. But let’s keep going anyway, to see what we’ve got. We know how to expand the right-hand terms in negative powers of n up to, say, O(n5): In n- = -n-l + Lgp2 - in-3 -+ $np4 + o(ne5) ; n+l 1 -= n -1 _ ,-2 + np3 - - n+ + O(np5); n+l 1 ___ zz (n-t 1)2 nm2 - 2n ~3 + 3n m4 + O(np5) ; np4 + O(n5) Therefore the terms on the right of our approximation add up to ln2 + t + & - j$ + (-1 - t + t)n-’ + (+ - t - & + &)np2 +(-+-&)n-3 + (1; - i + & + & - &)nm4 + R4(n) = ln2+$$-n-‘+R4(n)+O(n~5). The coefficients of ne2, np3, and nm4 cancel nicely, as they should. If all were well with the world, we would be able to show that R,+(n) is asymptotically small, maybe O(n5), and we would have an approximation to the sum. But we can’t possibly show this, because we happen to know that the correct constant term is 1, not ln2 + s (which is approximately 0.9978). So R4(n) is actually equal to G - ln2 + O(n4), but Euler’s summation formula doesn’t tell us this. In other words, we lose. One way to try fixing things is to notice that the constant terms in the approximation form a. pattern, if we let m get larger and larger: ln2-tB,+t.~B2-~.~B3+~.~B4--.~B5+... Perhaps we can show that this series approaches 1 as the number of terms becomes infinite? But no; the Bernoulli numbers get very large. For example, f322 = @$$ > 6192; therefore IR22(n)l will be much larger than 1 R4 (n) 1. We lose totally. There is a way out, however, and this escape route will turn out to be important in other applications of Euler’s formula. The key is to notice that R4(n) approaches a definite limit as n + 00: lim R4(n) = -jyB4({x))(&&) dx = R4(m) n-02
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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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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
Page 426 and 427: 412 DISCRETE PROBABILITY to be a pg
Page 428 and 429: 414 DISCRETE PROBABILITY 10 What’
Page 430 and 431: 416 DISCRETE PROBABILITY 28 What is
Page 432 and 433: 418 DISCRETE PROBABILITY 38 What is
Page 434 and 435: 420 DISCRETE PROBABIL1T.Y a Once he
Page 436 and 437: 422 DISCRETE PROBABILITY 50 Conside
Page 438 and 439: 424 DISCRETE PROBABILITY 58 Are the
Page 440 and 441: 426 ASYMPTOTICS The word asymptotic
Page 442 and 443: 428 ASYMPTOTIC3 How about the funct
Page 444 and 445: 430 ASYMPTOTICS N. G. de Bruijn beg
Page 446 and 447: 432 ASYMPTOTICS subtle consideratio
Page 448 and 449: 434 ASYMPTOTICS inefficient “beca
Page 450 and 451: 436 ASYMPTOTICS 9.3 0 MANIPULATION
Page 452 and 453: 138 ASYMPTOTICS Table 438 Asymptoti
Page 454 and 455: 440 ASYMPTOTICS Notice how the 0 te
Page 456 and 457: 442 ASYMPTOTICS Problem 3: The nth
Page 458 and 459: 444 ASYMPTOTICS Problem 4: A sum fr
Page 460 and 461: 446 ASYMPTOTICS Problem 5: An infin
Page 462 and 463: 448 ASYMPTOTICS We might as well do
Page 464 and 465: 450 ASYMPTOTIC3 a rough estimate an
Page 466 and 467: 452 ASYMPTOTICS This last estimate
Page 468 and 469: 454 ASYMPTOTICS Here’s another ex
Page 470 and 471: 456 ASYMPTOTICS On the left is a ty
Page 472 and 473: 458 ASYMPTOTICS (The Bernoulli poly
Page 474 and 475: 460 ASYMPTOTICS The graph of B,(x)
Page 478 and 479: 464 ASYMPTOTICS The integral JT B4(
Page 480 and 481: 466 ASYMPTOTIC3 where 0 < a,,,, < 1
Page 482 and 483: 468 ASYMPTOTICS In Chapter 5, equat
Page 484 and 485: 470 ASYMPTOTICS For example, we hav
Page 486 and 487: 472 ASYMPTOTICS The final task seem
Page 488 and 489: 474 ASYMPTOTICS This is our approxi
Page 490 and 491: 476 ASYMPTOTIC23 9 Prove (9.22) rig
Page 492 and 493: 478 ASYMPTOTICS 39 Evaluate xOsk
Page 494 and 495: 480 ASYMPTOTICS a What is the asymp
Page 496 and 497: 482 ASYMPTOTIC3 Research problems 6
Page 498 and 499: 484 ANSWERS TO EXERCISES Venn [294]
Page 500 and 501: 486 ANSWERS TO EXERCISES move the b
Page 502 and 503: 488 ANSWERS TO EXERCISES 2.2 This i
Page 504 and 505: 490 ANSWERS TO EXERCISES 2.24 Summi
Page 506 and 507: 492 ANSWERS TO EXERCISES 2.36 (a) B
Page 508 and 509: 494 ANSWERS TO EXERCISES 3.21 If 10
Page 510 and 511: 496 ANSWERS TO EXERCISES the circle
Page 512 and 513: 498 ANSWERS TO EXERCISES 3.40 Let L
Page 514 and 515: 500 ANSWERS TO EXERCISES 3.50 H. S.
Page 516 and 517: 502 ANSWERS TO EXERCISES 4.22 (b,
Page 518 and 519: 504 ANSWERS TO EXERCISES 4.40 Let f
Page 520 and 521: 506 ANSWERS TO EXERCISES 4.50 (a) I
Page 522 and 523: 508 ANSWERS TO EXERCISES and (k- 1)
Page 524 and 525: 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
Page 598 and 599:
584 BIBLIOGRAPHY 78 79 80 81 82 83
Page 600 and 601:
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
Page 618 and 619:
604 CREDITS FOR EXERCISES 6.46 6.47
Page 620 and 621:
Index WHEN AN INDEX ENTRY refers to
Page 622 and 623:
608 INDEX Bois-Reymond, Paul David
Page 624 and 625:
610 INDEX Degenerate hypergeometric
Page 626 and 627:
612 INDEX Feder, Tom&s, 604. Feigen
Page 628 and 629:
614 INDEX Hall, Marshall, Jr., 588.
Page 630 and 631:
616 INDEX Lincoln, Abraham, 387. Li
Page 632 and 633:
618 INDEX Patashnik, Oren, iii, iv,
Page 634 and 635:
620 INDEX Running time, 411-412. Ru
Page 636 and 637:
622 INDEX Sylvester, James Joseph,
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List of Tables Sums and differences
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