The Art and Craft of Problem Solving

9.4 POWER SERIES AND EULERIAN MATHEMATICS 353

The number of heads depends purely on the relative ranking of A 2

among the five random numbers.

But since these numbers are all uniformly distributed, the rank of anyone number

is also uniformly distributed; i.e., it is just as likely that A2 is the largest as it is the

smallest as it is the 3rd largest, etc. So the probabilities for all five outcomes are the

same, namely, 1/5.

Integration and generatingfunctionology obscured a relatively simple situation.

The probabilities were uniform because the numbers were uniform, and thus their

rankings were uniform.

The underlying principle, the "why" that explains this problem, is no stranger to

you. It has explained so much in this book, so it is fitting that it is the word we end

with: Symmetry. _

Problems and Exercises

9.4.10 Verify that the sum in Example 9.4.1 does not

converge uniformly (look at what happens near x = 0).

9.4. 11 Consider the series

I +x+x-+x^2 3 + ... = --I ,

I-x

which converges for Ixl < I.

(a) Show that this series does not converge uni­

formly.

(b) Show that this series does converge uniformly

for Ixl S 0.9999.

9.4. 12 Prove an important generalization of the bino­

mial theorem, which states that

where

(a).

. =
a(a-I)···a- r+ I
,
r r!
r=I,2,3,. ...

Notice that this definition of binomial coefficient

agrees with the combinatorial one that is defined only

for positive integral a. Also note that the series above

will terminate if a is a positive integer. Discuss con­

vergence. Does it depend on a?

9.4. 13 (Putnam 1992) Define C( a) to be the coef­

ficient of xl^992 in the power series about x = 0 of

(I +x)a. Evaluate

10

1

(C( -y -I) j�

y

� k ) dy.

9.4. 14 Prove the assertion on page 346 that concluded

with (9), which stated that the second-degree Taylor

polynomial is the "best" quadratic approximation to a

function.

9.4. 15 Use power series to prove that e'+Y = e'eY.

9.4.16 (Putnam 1998) Let N be the positive integer

with 1998 decimal digits, all of them I; that is,

N=IIII···ll.

Find the thousandth digit after the decimal point of

VN.

9.4. 17 Let x> 1. Evaluate the sum

x x^2

--+ -,----.,--,----=--

x+ 1 (x + l)(x^2 + I)

x^4

+

(x+ 1)(x^2 + 1)(x 4 + I)

+ ....

9.4. 18 (Putnam 1990) Prove that for Ixl < I, Izl > I,

00

1 + }; (I +xj)Pj = 0,

J=I

where Pj is

(l-z)(l-zx)(l-zx^2 )···(I- zxj-l)

(z -x)(z -x^2 )(z -x3) ... (z -xj)