Chapter 10

What happens if you add a finite number of terms to a divergent series or delete a finite number of terms from a divergent series? Give reasons for your answer.

If \(\sum a_{n}\) converges and \(\sum b_{n}\) diverges, can anything be said about their term-by-term sum \(\sum\left(a_{n}+b_{n}\right) ?\) Give reasons for your answer.

Make up a geometric series \(\sum a r^{n-1}\) that converges to the number 5 if a. a=2 b. a=13 / 2

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(27-90\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\frac{1}{n} \int_{1}^{n} \frac{1}{x} d x $$

Find the value of \(b\) for which $$ 1+e^{b}+e^{2 b}+e^{3 b}+\cdots=9 $$

For what values of \(r\) does the infinite series $$ 1+2 r+r^{2}+2 r^{3}+r^{4}+2 r^{5}+r^{6}+\cdots $$ converge? Find the sum of the series when it converges.

In Exercises \(91-98\) , assume that each sequence converges and find its limit. $$ a_{1}=2, \quad a_{n+1}=\frac{72}{1+a_{n}} $$

In Exercises \(91-98\) , assume that each sequence converges and find its limit. $$ a_{1}=-1, \quad a_{n+1}=\frac{a_{n}+6}{a_{n}+2} $$

Drug dosage A patient takes a 300 mg tablet for the control of high blood pressure every morning at the same time. The concentration of the drug in the patient's system decays exponentially at a constant hourly rate of \(k=0.12\) . a. How many milligrams of the drug are in the patient's system just before the second tablet is taken? Just before the third tablet is taken? b. In the long run, after taking the medication for at least six months, what quantity of drug is in the patient's body just before taking the next regularly scheduled morning tablet?

In Exercises \(91-98\) , assume that each sequence converges and find its limit. $$ a_{1}=-4, \quad a_{n+1}=\sqrt{8+2 a_{n}} $$

Show that the error \(\left(L-s_{n}\right)\) obtained by replacing a convergen geometric series with one of its partial sums \(s_{n}\) is \(a r^{n} /(1-r)\) .

In Exercises \(91-98\) , assume that each sequence converges and find its limit. $$ a_{1}=0, \quad a_{n+1}=\sqrt{8+2 a_{n}} $$

The Cantor set To construct this set, we begin with the closed interval \([0,1]\) . From that interval, remove the middle open interval \((1 / 3,2 / 3),\) leaving the two closed intervals \([0,1 / 3]\) and \([2 / 3,1] .\) At the second step we remove the open middle third interval from each of those remaining. From \([0,1 / 3]\) we remove the open interval \((1 / 9,2 / 9),\) and from \([2 / 3,1]\) we remove \((7 / 9,8 / 9),\) leaving behind the four closed intervals \([0,1 / 9]\) \([2 / 9,1 / 3],[2 / 3,7 / 9],\) and \([8 / 9,1] .\) At the next step, we remove the middle open third interval from each closed interval left behind, so \((1 / 27,2 / 27)\) is removed from \([0,1 / 9],\) leaving the closed intervals \([0,1 / 27]\) and \([2 / 27,1 / 9] ;(7 / 27,8 / 27)\) is removed from \([2 / 9,1 / 3],\) leaving behind \([2 / 9,7 / 27]\) and \([8 / 27,1 / 3],\) and so forth. We continue this process repeatedly without stopping, at each step removing the open third interval from every closed interval remaining behind from the preceding step. The numbers remaining in the interval \([0,1],\) after all open middle third intervals have been removed, are the points in the Cantor set (named after Georg Cantor, \(1845-1918\) . The set has some interesting properties. a. The Cantor set contains infinitely many numbers in \([0,1]\) . List 12 numbers that belong to the Cantor set. b. Show, by summing an appropriate geometric series, that the total length of all the open middle third intervals that have been removed from \([0,1]\) is equal to \(1 .\)

In Exercises \(91-98\) , assume that each sequence converges and find its limit. $$ a_{1}=5, \quad a_{n+1}=\sqrt{5 a_{n}} $$

Helga von Koch's snowflake curve Helga von Koch's snowflake is a curve of infinite length that encloses a region of finite area. To see why this is so, suppose the curve is generated by starting with an equilateral triangle whose sides have length 1. a. Find the length \(L_{n}\) of the \(n\) th curve \(C_{n}\) and show that \(\quad \lim _{n \rightarrow \infty} L_{n}=\infty\) b. Find the area \(A_{n}\) of the region enclosed by \(C_{n}\) and show that \(\quad \lim _{n \rightarrow \infty} A_{n}=(8 / 5) A_{1} .\)

In Exercises \(91-98\) , assume that each sequence converges and find its limit. $$ a_{1}=3, \quad a_{n+1}=12-\sqrt{a_{n}} $$

In Exercises \(91-98\) , assume that each sequence converges and find its limit. $$ 2,2+\frac{1}{2}, 2+\frac{1}{2+\frac{1}{2}}, 2+\frac{1}{2+\frac{1}{2+\frac{1}{2}}}, \dots $$

In Exercises \(91-98\) , assume that each sequence converges and find its limit. $$ \begin{array}{l}{\sqrt{1}, \sqrt{1+\sqrt{1}}, \sqrt{1+\sqrt{1+\sqrt{1}}}} \\\ {\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1}}}}, \ldots}\end{array} $$

The first term of a sequence is \(x_{1}=1 .\) Each succeeding term is the sum of all those that come before it: $$x_{n+1}=x_{1}+x_{2}+\dots+x_{n}$$ Write out enough early terms of the sequence to deduce a general formula for \(x_{n}\) that holds for \(n \geq 2\) .

A sequence of rational numbers is described as follows: $$\frac{1}{1}, \frac{3}{2}, \frac{7}{5}, \frac{17}{12}, \ldots, \frac{a}{b}, \frac{a+2 b}{a+b}, \ldots$$ Here the numerators form one sequence, the denominators form a second sequence, and their ratios form a third sequence. Let \(x_{n}\) and \(y_{n}\) be, respectively, the numerator and the denominator of the \(n\) th fraction \(r_{n}=x_{n} / y_{n}\) a. Verify that \(x_{1}^{2}-2 y_{1}^{2}=-1, x_{2}^{2}-2 y_{2}^{2}=+1\) and, more generally, that if \(a^{2}-2 b^{2}=-1\) or \(+1,\) then $$(a+2 b)^{2}-2(a+b)^{2}=+1 \quad\( or \)\quad-1$$ respectively. b. The fractions \(r_{n}=x_{n} / y_{n}\) approach a limit as \(n\) increases. What is that limit? (Hint: Use part (a) to show that \(r_{n}^{2}-2=\pm\left(1 / y_{n}\right)^{2}\) and that \(y_{n}\) is not less than \(n .\) )

Newton's method The following sequences come from the recursion formula for Newton's method, $$x_{n+1}=x_{n}-\frac{f\left(x_{n}\right)}{f^{\prime}\left(x_{n}\right)}$$ Do the sequences converge? If so, to what value? In each case, begin by identifying the function \(f\) that generates the sequence. $$ \begin{array}{l}{\text { a. } x_{0}=1, \quad x_{n+1}=x_{n}-\frac{x_{n}^{2}-2}{2 x_{n}}=\frac{x_{n}}{2}+\frac{1}{x_{n}}} \\\ {\text { b. } x_{0}=1, \quad x_{n+1}=x_{n}-\frac{\tan x_{n}-1}{\sec ^{2} x_{n}}} \\ {\text { c. } x_{0}=1, \quad x_{n+1}=x_{n}-1}\end{array} $$

a. Suppose that \(f(x)\) is differentiable for all \(x\) in \([0,1]\) and that \(f(0)=0 .\) Define sequence \(\left\\{a_{n}\right\\}\) by the rule \(a_{n}=n f(1 / n)\) . Show that \(\lim _{n \rightarrow \infty} a_{n}=f^{\prime}(0) .\) Use the result in part (a) to find the limits of the following sequences \(\left\\{a_{n}\right\\} .\) $$\begin{aligned} \text { b. } a_{n} &=n \tan ^{-1} \frac{1}{n} & \text { c. } a_{n}=n\left(e^{1 / n}-1\right) \\ \text { d. } a_{n} &=n \ln \left(1+\frac{2}{n}\right) \end{aligned}$$

Pythagorean triples A triple of positive integers \(a, b,\) and \(c\) is called a Pythagorean triple if \(a^{2}+b^{2}=c^{2} .\) Let \(a\) be an odd positive integer and let $$ b=\left\lfloor\frac{a^{2}}{2}\right\rfloor \text { and } c=\left\lceil\frac{a^{2}}{2}\right\rceil $$ be, respectively, the integer floor and ceiling for \(a^{2} / 2\) a. Show that \(a^{2}+b^{2}=c^{2} .\) (Hint: Let \(a=2 n+1\) and express \(b\) and \(c\) in terms of \(n .\) b. By direct calculation, or by appealing to the accompanying figure, find $$ \lim _{a \rightarrow \infty} \frac{\left\lfloor\frac{a^{2}}{2}\right\rfloor}{ | \frac{a^{2}}{2} \rceil} $$

Prove that \(\lim _{n \rightarrow \infty} \sqrt[n]{n}=1\)

$$ \lim _{n \rightarrow \infty} x^{1 / n}=1,(x>0) $$

In Exercises \(111-114,\) determine if the sequence is monotonic and if it is bounded. $$a_{n}=\frac{3 n+1}{n+1}$$

In Exercises \(111-114,\) determine if the sequence is monotonic and if it is bounded. $$ a_{n}=\frac{(2 n+3) !}{(n+1) !} $$

In Exercises \(111-114,\) determine if the sequence is monotonic and if it is bounded. $$ a_{n}=\frac{2^{n} 3^{n}}{n !} $$

In Exercises \(111-114,\) determine if the sequence is monotonic and if it is bounded. $$ a_{n}=2-\frac{2}{n}-\frac{1}{2^{n}} $$

Which of the sequences in Exercises \(115-124\) converge, and which diverge? Give reasons for your answers. $$ a_{n}=n-\frac{1}{n} $$

Which of the sequences in Exercises \(115-124\) converge, and which diverge? Give reasons for your answers. $$ a_{n}=\left((-1)^{n}+1\right)\left(\frac{n+1}{n}\right) $$

In Exercises \(125-126,\) use the definition of convergence to prove the given limit. $$\lim _{n \rightarrow \infty} \frac{\sin n}{n}=0$$

In Exercises \(125-126,\) use the definition of convergence to prove the given limit. $$\lim _{n \rightarrow \infty}\left(1-\frac{1}{n^{2}}\right)=1$$

The sequence \(\\{n /(n+1)\\}\) has a least upper bound of 1 Show that if \(M\) is a number less than \(1,\) then the terms of \(\\{n /(n+1)\\}\) eventually exceed \(M .\) That is, if \(M<1\) there is an integer \(N\) such that \(n /(n+1)>M\) whenever \(n>N .\) since \(n /(n+1)<1\) for every \(n,\) this proves that 1 is a least upper bound for \(\\{n /(n+1)\\} .\)

Uniqueness of least upper bounds Show that if \(M_{1}\) and \(M_{2}\) are least upper bounds for the sequence \(\left\\{a_{n}\right\\},\) then \(M_{1}=M_{2}\) That is, a sequence cannot have two different least upper bounds.

Prove that if \(\left\\{a_{n}\right\\}\) is a convergent sequence, then to every positive number \(\epsilon\) there corresponds an integer \(N\) such that for all \(m\) and \(n,\) $$ m>N \quad \text { and } \quad n>N \Rightarrow\left|a_{m}-a_{n}\right|<\epsilon $$

Uniqueness of limits Prove that limits of sequences are unique. That is, show that if \(L_{1}\) and \(L_{2}\) are numbers such that \(a_{n} \rightarrow L_{1}\) and \(a_{n} \rightarrow L_{2},\) then \(L_{1}=L_{2}\) .

Limits and subsequences If the terms of one sequence appear in another sequence in their given order, we call the first sequence a subsequence of the second. Prove that if two sub- sequences of a sequence \(\left\\{a_{n}\right\\}\) have different limits \(L_{1} \neq L_{2}\) then \(\left\\{a_{n}\right\\}\) diverges.

For a sequence \(\left\\{a_{n}\right\\}\) the terms of even index are denoted by \(a_{2 k}\) and the terms of odd index by \(a_{2 k+1} .\) Prove that if \(a_{2 k} \rightarrow L\) and \(a_{2 k+1} \rightarrow L,\) then \(a_{n} \rightarrow L\)

Prove that a sequence \(\left\\{a_{n}\right\\}\) converges to 0 if and only if the sequence of absolute values \(\left\\{\left|a_{n}\right|\right\\}\) converges to \(0 .\)

Sequences generated by Newton's method Newton's method, applied to a differentiable function \(f(x)\) , begins with a starting value \(x_{0}\) and constructs from it a sequence of numbers \(\left\\{x_{n}\right\\}\) that under favorable circumstances converges to a zero of \(f .\) The recursion formula for the sequence is $$x_{n+1}=x_{n}-\frac{f\left(x_{n}\right)}{f^{\prime}\left(x_{n}\right)}$$ a. Show that the recursion formula for \(f(x)=x^{2}-a, a>0\) can be written as \(x_{n+1}=\left(x_{n}+a / x_{n}\right) / 2 .\) b. Starting with \(x_{0}=1\) and \(a=3,\) calculate successive terms of the sequence until the display begins to repeat. What number is being approximated? Explain.

Use a CAS to perform the following steps for the sequences in Exercises \(137-148 .\) a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit L? b. If the sequence converges, find an integer \(N\) such that \(\quad\left|a_{n}-L\right| \leq 0.01\) for \(n \geq N .\) How far in the sequence do you have to get for the terms to lie within 0.0001 of \(L ?\) $$a_{n}=\sqrt[n]{n}$$

Use a CAS to perform the following steps for the sequences in Exercises \(137-148 .\) a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit L? b. If the sequence converges, find an integer \(N\) such that \(\quad\left|a_{n}-L\right| \leq 0.01\) for \(n \geq N .\) How far in the sequence do you have to get for the terms to lie within 0.0001 of \(L ?\) $$ a_{n}=\left(1+\frac{0.5}{n}\right)^{n} $$

Use a CAS to perform the following steps for the sequences in Exercises \(137-148 .\) a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit L? b. If the sequence converges, find an integer \(N\) such that \(\quad\left|a_{n}-L\right| \leq 0.01\) for \(n \geq N .\) How far in the sequence do you have to get for the terms to lie within 0.0001 of \(L ?\) $$ a_{1}=1, \quad a_{n+1}=a_{n}+\frac{1}{5^{n}} $$

Use a CAS to perform the following steps for the sequences in Exercises \(137-148 .\) a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit L? b. If the sequence converges, find an integer \(N\) such that \(\quad\left|a_{n}-L\right| \leq 0.01\) for \(n \geq N .\) How far in the sequence do you have to get for the terms to lie within 0.0001 of \(L ?\) $$ a_{1}=1, \quad a_{n+1}=a_{n}+(-2)^{n} $$

Use a CAS to perform the following steps for the sequences in Exercises \(137-148 .\) a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit L? b. If the sequence converges, find an integer \(N\) such that \(\quad\left|a_{n}-L\right| \leq 0.01\) for \(n \geq N .\) How far in the sequence do you have to get for the terms to lie within 0.0001 of \(L ?\) $$ a_{n}=\sin n $$

Use a CAS to perform the following steps for the sequences in Exercises \(137-148 .\) a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit L? b. If the sequence converges, find an integer \(N\) such that \(\quad\left|a_{n}-L\right| \leq 0.01\) for \(n \geq N .\) How far in the sequence do you have to get for the terms to lie within 0.0001 of \(L ?\) $$ a_{n}=n \sin \frac{1}{n} $$

Use a CAS to perform the following steps for the sequences in Exercises \(137-148 .\) a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit L? b. If the sequence converges, find an integer \(N\) such that \(\quad\left|a_{n}-L\right| \leq 0.01\) for \(n \geq N .\) How far in the sequence do you have to get for the terms to lie within 0.0001 of \(L ?\) $$ a_{n}=\frac{\sin n}{n} $$

Use a CAS to perform the following steps for the sequences in Exercises \(137-148 .\) a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit L? b. If the sequence converges, find an integer \(N\) such that \(\quad\left|a_{n}-L\right| \leq 0.01\) for \(n \geq N .\) How far in the sequence do you have to get for the terms to lie within 0.0001 of \(L ?\) $$ a_{n}=\frac{\ln n}{n} $$

Use a CAS to perform the following steps for the sequences in Exercises \(137-148 .\) a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit L? b. If the sequence converges, find an integer \(N\) such that \(\quad\left|a_{n}-L\right| \leq 0.01\) for \(n \geq N .\) How far in the sequence do you have to get for the terms to lie within 0.0001 of \(L ?\) $$ a_{n}=(0.9999)^{n} $$