The Art and Craft of Problem Solving

344 CHAPTER 9 CALCULUS If fn(x) ---t f(x) unifonnly, then J fn (x)dx ---t J f(x)dx. Unifonnly convergent power series can be di ...

9.4 POWER SERIES AND EULERIAN MATHEMATICS 345 for all tED. Integrating this with respect to t from a to x yields^12 M 3 (x-a):: ...

346 CHAPTER 9 CALCULUS other words, complete "local" information yields complete global information! This is worth pondering. In ...

9.4 POWER SERIES AND EULERIAN MATHEMATICS 347 The early stages of any investigation, The early stages of any person's mathemati ...

348 CHAPTER 9 CALCULUS Determine, with proof, the largest number that is the product of positive integers whose sum is 197 6. Pa ...

9.4 POWER SERIES AND EULERIAN MATHEMATICS 349 appear exactly once, because prime factorization is unique (see the discussion of ...

350 CHAPTER 9 CALCULUS won't work; the constant term is infinite, which is horribly wrong. The power series (10) tells us that t ...

9.4 POWER SERIES AND EULERIAN MATHEMATICS 351 However, p was randomly generated, so the correct probability is the average-i.e., ...

352 CHAPTER 9 CALCULUS Thus we have f(x) = 101 (1-p+ pxtdp = (I -p + px )n+ I ] I (n+l)(x -l) 0 I (�+I -I) = n+1 x-I I (_J! n-I ...

9.4 POWER SERIES AND EULERIAN MATHEMATICS 353 The number of heads depends purely on the relative ranking of A 2 among the five r ...

354 CHAPTER 9 CALCULUS 9.4. 19 (Putnam 1997) Evaluate r[(x_X^3 +�_�+ ... ) Jo^2 2·4^2 ·4·^6 x (^1 + �� + 2 t 42 + 22. �. 62 + .. ...

9.4 POWER SERIES AND EULERIAN MATHEMATICS 355 9.4.37 Analogous to the concept of perfect numbers (see Problems 7.5.33-7.5.35) ar ...

References and Further Reading Further Reading 356 Here are a few suggestions for expanding your horizons. This is by no means a ...

References REFERENCES AND FURTHER READING 357 text [15] is a fine alternative. For more advanced reading, there are three books ...

358 REFERENCES AND FURTHER READING 12. Martin Gardner. Wheels, Life and Other Mathematical Amusements. Freeman, 19 83. 13. Georg ...

REFERENCES AND FURTHER READING 359 34. George P61ya, Robert E. Tarjan, and Donald R. Woods. Notes on Introductory Combinatorics. ...

Index AAS condition, 260 absolute value, 51, 174 of a complex number, 121 of terms in a series, 345 of terms of a sequence, 319 ...

chord, 264 andradii, 264 Chvatal, Vaclav, 55 circles and arcs, 265 arc, 264 chord, 264 general, 264-266 generalized, 308 inscrib ...

362 INDEX rules for, 94, 107 division algorithm for integers, 83, 224, 226 for polynomials, 164 divisors common, 60 number of, 3 ...

inequalities AM-OM, see AM-OM inequality Bernoulli's, 51, 330 Cauchy-Schwan, see Cauchy- Schwan inequality Chebyshev, 185 Euler' ...