Answer

On p. xiii, 6 lines from the bottom, my Web address is now changed to www.mathsci.appstate.edu/~wak.

On p. xiii, add “Numerical Integration” as the last line.

On p. xv, in line 4 of paragraph 2, change “Este” to “Esty”.

On p. 3, in the last line of the footnote, it should be “Section 1.6”.

On p. 4, in the answer to Example 1.1.4, “” should be “”.

On p. 4, in line above Theorem 1.1.5, the word "or" should be "and".

On p. 8, in Example 1.2.2, in the definition of , should be changed to .

On p. 11, line above Remark 1.2.8, “properly” should be changed to “property”.

On p. 43, line (A6) should be: “There exists an element in , distinct from 0, which we denote … .”

On p. 47, in Exercise 2(c), change “” to a “”.

On p. 51, line (b) of Corollary 1.8.6 should be: “”.

On p. 53, in Exercise 21(a), change the last inequality to a multiplication.

On p. 75, insert a minus sign into next to the last line in the Proof of part (a).

On p. 77, last term in the next to the last line should be times the square root instead of th root.

On p. 78, in line 7 of Remark 2.2.5, “” should be changed to “”.

On p. 83, change the text that comes above Theorem 2.3.6 to the following.

Proof. In determining whether to consider or , writing out a few terms or simply observing that one of the leading terms has a negative coefficient and the other leading coefficient is positive, suggests that the limit is . Let be given. We want to find so that for all , we will have . But, solving for is not easy. To avoid this task, we need to bound above by something that tends to . Hence, in this problem we need to make the numerator a larger negative expression, and the denominator a smaller positive expression. Although there are many different choices, let us write

, for , and , for .

Therefore, picking , since the numerator is negative, we write

But yields . Thus, if , for all , we have .

The preceding lengthy proof can be shortened as shown next. Hopefully, this “behind-the-scenes” proof provided insight.

Pick any . Let . If , we have

Hence, . It should be noted that perhaps showing that tends to and implementing part (d) of Theorem 2.3.3 would be an easier approach. Moreover, since , using the comparison test would also prove the divergence to .

See Exercises 3 and 9 for more information concerning rational expressions. There are other ways to determine divergence to infinity. The next result relates ideas from previous sections to the divergence to infinity.

On p. 84, change part (c) of Theorem 2.3.7 to “If , then may converge, diverge to plus or minus infinity, or oscillate.”

On p. 85, in top line, change “three” to “four”.

On p. 85, second line of the proof just below Example 2.3.8, change “” to “”.

On p. 86, Exercise 8, change “diverges to infinity” to “diverges to plus or minus infinity”.

On p. 87, Exercise 9, change “diverges to infinity” to “diverges to plus or minus infinity”.

On p. 87, in Exercise 15 “three” to “four”.

On p. 87, Exercise 16, change “diverge to infinity” to “diverge to plus or minus infinity”.

On p. 89, line (d) of Definition 2.4.1, should have “” instead of “”.

On p. 103, Exercise 1, delete the first sentence. Start the problem with “Prove that …”.

On p. 109, 5^th line from the top change the sentences “The existence… Section 2.5. Why?” to “In Exercise 11 we are asked to show that there is a subsequence of that converges to .”

On p. 110, in Exercise 2(c), change “” to “ or ”.

On p. 110, Exercise 5 should read as follows: Prove that every unbounded above sequence contains a monotone subsequence that diverges to plus infinity.

On p. 111 add Exercise 11 which states: “Complete the proof of Theorem 2.6.4.”

On p. 120, part (b) should read as “ must be eventually bounded … .”

On p. 122, in Remark 3.1.12, change both to where .

On p. 125, in Exercise 14, prove the given two limits without using Theorem 3.1.13(b).

On p. 126, in Figure 3.2.1, fill in the point and make the vertical line above dotted.

On p. 125, in line right above Example 3.2.10, change “Theorem 3.2.5” to “Theorem 3.2.6”.

On p. 168, Exercise 17 should read as “Give an example of a function that is a continuous injection… .”

On p. 171, top line, change “what” to “that”.

On p. 174, Exercise 3 should be changed to:

(a) Prove Theorem 4.4.7.

(b) Suppose is continuous. Prove that if and are both finite, then is bounded on . Explain why the converse is not true.

(d) Use Corollary 4.4.8 to prove that is not uniformly continuous on but is uniformly continuous on .

On p. 184, in Definition 5.1.1, second line, replace “” by “ is continuous at ”.

On p. 191, parts (a) and (b) of Exercise should be changed to:

(a) continuous at exactly one point and differentiable at exactly one point.

(b) continuous at exactly two points and differentiable at exactly two points.

On p. 197, in the second line of the proof, delete “Thus, or on ”.

On p. 197, next to the last line a derivative symbol is missing on .

On p. 198, in line 7 of the proof of Theorem 5.2.9, change “” to “”.

On p. 200, Exercise 7 should read as follows: “Give an example of a function that is differentiable at such that , but yet attains a relative extremum at .”

On p. 200, Exercise 8 should read as follows: “Give an example of a function that is continuous at , not differentiable at , but yet attains a relative extremum at .”

On p. 205, second indented equation has equal sign missing.

On p. 215, in the first line after the proof of Taylor’s theorem, add a word “ often” before the word “become.”

On p. 247, the last line of the proof of Theorem 6.2.1 should be “Since and are within of each other, so must be the upper and the lower integrals, proving the desired result.”

On p. 257, keep the first 5 lines of the proof of Theorem 6.4.2. The rest of the proof should be changed to what follows.

This is a Riemann sum and thus, it follows that . Since is an arbitrary partition, we have that

Lastly, since is Riemann integrable on , upper and lower integrals must be equal and hence, .

On p. 261, Exercise 1, add at the end of the line “with ”.

On p. 295, change the top of page to what follows.

For any sequence , we can define a related sequence , where

Thus, is the sum up to the term . The sequence is called the sequence of partial sums of the series . (See Exercise 15 of Section 2.2.) Subscripts are dummy variables. …

On p. 295, in Definition 7.1.2 and in Remark 7.1.3, change all to .

On p. 296, in the “Answer” 4 lines from the bottom, remove the first equality sign.

On p. 305, in Theorem 7.2.4(b), change “” to “”.

On p. 305, in the third line of the proof of part (a) of Theorem 7.2.4, this is the indented equation, is missing before ≤ sign.

On p. 307, in part (b) of Remark 7.2.8, in the first line change ≥ to >.

On p. 313, in part (e), change “both ratio tests” to “Theorem 7.3.3 and Corollary 7.3.5”.

On p. 323, Exercise 5 should start with three additional words “For each part,”.

On p. 344, in the Proof of part (b), second sentence should be “Thus, choose a sequence in the interval that converges to 1, say, .”

On p. 350, in Exercise 2 add at the end “(Do not use Theorem 8.3.4.)”

On p. 350, in Exercise 4 add at the end “for the increasing case.”

On p. 350, in Exercise 5(c), change “” to “ (or )”.

On p. 352, in the first line change “A series …” to “A converging series …”.

On p. 358, in Exercise 11(e), an equal sign is missing.

On p. 362, in part (b) of Theorem 8.5.8, change < to ≤.

On p. 454, in Exercise 1(a), change “top of a sphere” to “top half of a sphere”.

On p. 459, in the first line of the proof of Theorem 10.4.5, change “we need” to “it is sufficient”.

On p. 472, last line of the footnote should be: “See Part 3 of Section 12.8 in … .”

On p. 479, add the following paragraph on top of page.

It should be noted that finding all functions for which boils down to solving separable differential equation . This was the content of Exercise 31(a) in Section 5.3.