The Art and Craft of Problem Solving

328 CHAPTER 9 CALCULUS

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9.2.34 (Putnam 1994) Let (an) be a sequence ofposi­

tive reals such that. for all n. an ::; a 2 n + a 2 n+ I. Prove

that the sum � an converges if and only if the product

00 00 n=O

that � an diverges. il (1 + an) converges.

n= 1

9.2.35 Let (an) be a sequence whose terms alter­

nate in sign. and whose terms decrease monoton­

ically to zero in absolute value. (For example.

9.2.38 Let (ak) be the sequence used as in Prob-

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lem 9.2.35. This problem showed that � an con­

00 n= 1

1,- 1/2,+1/3,- 1/4, .... ) Show that � an con-

n= 1

verges.

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9.2.36 Suppose that an > 0 for all n and � an di-

n=O

verges. Prove that it is possible for f � to ei-

n=O^1 + nan

ther converge or diverge. depending on the choice of

the sequence (an).

9.2.37 Sums and Products. Let an > 0 for all n. Prove

9.3 Differentiation and Integration

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verges. Notice also that � Ian I diverges (it is the har-

n= 1

monic series. after all). Use these two facts to show

that given any real number x. it is possible to rearrange

the terms of the sequence (an) so that the new sum

converges to x. By "rearrange" we mean reorder. For

example. one rearrangement would be the series

1 I 1 1

3"

+

19

• 100
  
  
  • liT
    -1 + ....
  
  
  
  

9.2.39 Can you improve the bounds found in Exam­

ple 9.2.1?

Approximation and Curve Sketching

You certainly know that the derivative f'{X) of the function f{x) has two interpreta­

tions: a dynamic definition as rate of change [of f{x) with respectto x], and a geometric

definition as slope of the tangent line to the graph of y = f{x) at the point (x, f{x)).

The rate-of-change definition is especially useful for understanding how functions

grow. More elaborate infonnation comes from the second derivative f"{x), which of

course measures how fast the derivative is changing. Sometimes just simple analysis

of the signs of f' and f" is enough to solve fairly complicated problems.

Example 9.3.1 Reread Example 2.2.7 on page 34, in which we studied the inequality

p{x) � p' (x ) for a polynomial function p. Recall that we reduced the original problem

to the following assertion:

Prove that if p{x) is a polynomial with even degree with positive lead­

ing coefficient, and p{x) -p"{x) � o for all real x, then p{x) � Ofor

all real x.

Solution: The hypothesis that p{x) has even degree with positive leading coeffi­

cient means that

lim p{x) = lim p{x) = +00;

x--+-oo x--++oo

therefore the minimum value of p{x) is finite (since p is a polynomial, it only "blows

up" as x ---+ ±oo). Now let us argue by contradiction, and assume that p{x) is negative

for some values of x. Let p{a) < 0 be the minimum value of the function. Recall