The Art and Craft of Problem Solving

342 CHAPTER 9 CALCULUS

where a> I.

9.3.44 The Schwarz Inequality. The Cauchy-Schwarz

inequality has many generalizations. Here is one for

integrals, known as the Schwarz inequality:

Let f(x),g (x) be nonnegative continuous

functions defined on the interval [a,bj.

Then

(lb f(x)g(X)dX)

2

� lb (f(x) )^2 dx lb (g(x) )^2 dx.

(a) First, examine the Cauchy-Schwarz inequality

(see page 182) to verify that the integral in­

equality above is a very plausible "version,"

since after all, integrals are "basically" sums.

(b) Proving the Schwarz inequality is another mat­

ter. Using the "integral-as-a-sum" idea is prob­

lematic, since limits are involved. However, we

presented two other alternate proofs of Cauchy-

Schwarz in Problems 5.5.35-5.5.36. Use one

(or both) of these to come up with a nice proof

of Schwarz's inequality.

(c) Generalize the inequality a tiny bit: remove the

hypothesis that f and g are nonnegative, and re­

place the conclusion with

(l

b If(x)g(x)ld

X)

2

� l

b

(f(x))^2 dx l

b

(g(x))^2 dx.

(d) Under what circumstances is Schwarz's in­

equality actually an equality?

9.3.45 (Turkey 1996) Given real numbers

O=XI <X 2 < ... < X 2 n <x 2 n+ 1 = I

with Xi+ 1 -Xi � h for I � i � 2 n, show that

I-h n l+h

• 2
  
  
  • < �X 2 i(X 2 i+ 1 -X 2 i-tl < --.
    1= 1 2
  
  
  
  

9.4 Power Series and Eulerian Mathematics

Don't Worry!

In this final section of the book, we take a brief look at infinite series whose terms

are not constants, but functions. This very quickly leads to technical questions of

convergence.

Example 9.4.1 Interpret the meaning of the infinite series

x^2 X^2 X^2

--+ + + ...

1 +x^2 (1 +x^2 )^2 (1 +x^2 )^3 '

where x can be any real number.

Solution: The only sensible interpretation is one that is consistent with our defi­

nition of series of constant terms. Thus we let

x^2

an(x) : =

(1 +x^2 )n'

n = 1 ,2, 3, ... ,

and define the function S(x) to be the limit of the partial sum functions. In other words,

if

then for each x E JR, we define

n

Sn(x) := }: ak(x),

k=1

S(x) := lim Sn(x),

n-->oo