Chapter 9

. Did you ever wonder how people find the decimal expansion of \(\pi\) to a large number of places? One method depends on the following identity (see Problem 34 of Section 1.9). $$ \pi=16 \tan ^{-1}\left(\frac{1}{5}\right)-4 \tan ^{-1}\left(\frac{1}{239}\right) $$ Find the first 6 digits of \(\pi\) using this identity and the series for \(\tan ^{-1} x\). (You will need terms through \(x^{9} / 9\) for \(\tan ^{-1}\left(\frac{1}{5}\right)\), but only the first term for \(\left.\tan ^{-1}(1 / 239) .\right)\) In 1706, John Machin used this method to calculate the first 100 digits of \(\pi\), while in 1973 , Jean Guilloud and Martine Bouyer found the first 1 million digits using the related identity $$ \pi=48 \tan ^{-1}\left(\frac{1}{18}\right)+32 \tan ^{-1}\left(\frac{1}{57}\right)-20 \tan ^{-1}\left(\frac{1}{239}\right) $$ In \(1983, \pi\) was calculated to over 16 million digits by a somewha different method. Of course, computers were used in these recen calculations.

Show that the alternating harmonic series $$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\cdots$$ (whose sum is actually \(\ln 2 \approx 0.69\) ) can be rearranged to converge to \(1.3\) by using the following steps. (a) Take enough of the positive terms \(1+\frac{1}{3}+\frac{1}{5}+\cdots\) to just exceed 1.3. (b) Now add enough of the negative terms \(-\frac{1}{2}-\frac{1}{4}-\frac{1}{6}-\cdots\) so that the partial sum \(S_{n}\) falls just below 1.3. (c) Add just enough more positive terms to again exceed 1.3, and so on.

In one version of Zeno's paradox, Achilles can run ten times as fast as the tortoise, but the tortoise has a 100 -yard headstart. Achilles cannot catch the tortoise, says Zeno, because when Achilles runs 100 yards the tortoise will have moved 10 yards ahead, when Achilles runs another 10 yards, the tortoise will have moved 1 yard ahead, and so on. Convince Zeno that Achilles will catch the tortoise and tell him exactly how many yards Achilles will have to run to do it.

Find the convergence set for each series. (a) \(\sum_{n=1}^{\infty} \frac{(3 x+1)^{n}}{n \cdot 2^{n}}\) (b) \(\sum_{n=1}^{\infty}(-1)^{n} \frac{(2 x-3)^{n}}{4^{n} \sqrt{n}}\)

Find a good bound for the maximum value of the given expression, given that \(c\) is in the stated interval. Answers may vary depending on the technique used. (See Example 5.) $$ \left|\frac{c^{2}-c}{\cos c}\right| ;\left[0, \frac{\pi}{4}\right] $$

Tom and Joel are good runners, both able to run at a constant speed of 10 miles per hour. Their amazing dog Trot can do even better; he runs at 20 miles per hour. Starting from towns 60 miles apart, Tom and Joel run toward each other while Trot runs back and forth between them. How far does Trot run by the time the boys meet? Assume that Trot started with Tom running toward Joel and that he is able to make instant turnarounds. Solve the problem two ways. (a) Use a geometric series. (b) Find a shorter way to do the problem.

Prove that if \(a_{n} \geq 0, b_{n}>0, \lim _{n \rightarrow \infty} a_{n} / b_{n}=0\), and \(\sum b_{n}\) converges then \(\sum a_{n}\) converges.

Suppose that Peter and Paul alternate tossing a coin for which the probability of a head is \(\frac{1}{3}\) and the probability of a tail is \(\frac{2}{3}\). If they toss until someone gets a head, and Peter goes first, what is the probability that Peter wins?

Suppose that \(a_{n+3}=a_{n}\) and let \(S(x)=\sum_{n=0} a_{n} x^{n} .\) Show that the series converges for \(|x|<1\) and give a formula for \(S(x)\).

Assuming that \(u_{1}=\sqrt{3}\) and \(u_{n+1}=\sqrt{3+u_{n}}\) determine a convergent sequence, find \(\lim _{n \rightarrow \infty} u_{n}\) to four decimal places.

Prove Theorem D as follows: Let $$ f(x)=1+\sum_{n=1}^{\infty}\left(\begin{array}{l} p \\ n \end{array}\right) x^{n} $$ (a) Show that the series converges for \(|x|<1\). (b) Show that \((1+x) f^{\prime}(x)=p f(x)\) and \(f(0)=1\). (c) Solve this differential equation to get \(f(x)=(1+x)^{p}\).

Prove that if \(a_{n} \geq 0, b_{n}>0, \lim _{n \rightarrow \infty} a_{n} / b_{n}=\infty\), and \(\Sigma b_{n}\) diverges then \(\sum a_{n}\) diverges.

Show that a conditionally convergent series can be rearranged so as to diverge.

Let $$ f(t)=\left\\{\begin{array}{ll} 0 & \text { if } t<0 \\ t^{4} & \text { if } t \geq 0 \end{array}\right. $$ Explain why \(f(t)\) cannot be represented by a Maclaurin series. Also show that, if \(g(t)\) gives the distance traveled by a car that is stationary for \(t<0\) and moving ahead for \(t \geq 0, g(t)\) cannot be represented by a Maclaurin series.

Suppose that \(\lim _{n \rightarrow \infty} n a_{n}=1 .\) Prove that \(\sum a_{n}\) diverges.

Show that \(\lim _{n \rightarrow \infty} a_{n}=0\) is not sufficient to guarantee the convergence of the alternating series \(\Sigma(-1)^{n+1} a_{n} .\) Hint: Alternate the terms of \(\sum 1 / n\) and \(\Sigma\left(-1 / n^{2}\right)\).

Suppose that Mary rolls a fair die until a "6" occurs. Let \(X\) denote the random variable that is the number of tosses needed for this " 6 " to occur. Find the probability distribution for \(X\) and verify that all the probabilities sum to 1 .

Find \(\lim _{n \rightarrow \infty} u_{n}\) of Problem 37 algebraically. Hint: Let \(u=\lim _{n \rightarrow \infty} u_{n} .\) Then, since \(u_{n+1}=\sqrt{3+u_{n}}, u=\sqrt{3+u}\). Now square both sides and solve for \(u\).

Let $$ f(x)=\left\\{\begin{array}{ll} e^{-1 / x^{2}} & \text { if } x \neq 0 \\ 0 & \text { if } x=0 \end{array}\right. $$ (a) Show that \(f^{\prime}(0)=0\) by using the definition of the derivative. (b) Show that \(f^{\prime \prime}(0)=0\). (c) Assuming the known fact that \(f^{(n)}(0)=0\) for all \(n\), find the Maclaurin series for \(f(x)\). (d) Does the Maclaurin series represent \(f(x) ?\) (e) When \(a=0\), the formula in Theorem \(\mathrm{B}\) is called Maclaurin's Formula. What is the remainder in Maclaurin's Formula for \(f(x)\) ? This shows that a Maclaurin series may exist and yet not represent the given function (the remainder does not tend to 0 as \(n \rightarrow \infty)\)

Prove that if \(\sum a_{n}\) is a convergent series of positive terms then \(\sum \ln \left(1+a_{n}\right)\) converges.

Discuss the convergence or divergence of $$ \begin{array}{r} \frac{1}{\sqrt{2}-1}-\frac{1}{\sqrt{2}+1}+\frac{1}{\sqrt{3}-1}-\frac{1}{\sqrt{3}+1}+ \\\ \frac{1}{\sqrt{4}-1}-\frac{1}{\sqrt{4}+1}+\cdots \end{array} $$

Prove that if \(\sum_{k=1}^{\infty} a_{k}^{2}\) and \(\sum_{k=1}^{\infty} b_{k}^{2}\) both converge then \(\sum_{k=1}^{\infty} a_{k} b_{k}\) converges absolutely. Hint: First show that \(2\left|a_{k} b_{k}\right| \leq a_{k}^{2}+b_{k}^{2}\).

$$ \text { Prove: If } \sum_{k=1}^{\infty} a_{k} \text { diverges, so does } \sum_{k=1}^{\infty} c a_{k} \text { for } c \neq 0 \text { . } $$

Assuming that \(u_{1}=0\) and \(u_{n+1}=1.1^{u_{n}}\) determine a convergent sequence, find \(\lim _{n \rightarrow \infty} u_{n}\) to four decimal places.

Test for convergence or divergence using the Root Test. (a) \(\sum_{n=2}^{\infty}\left(\frac{1}{\ln n}\right)^{n}\) (b) \(\sum_{n=1}^{\infty}\left(\frac{n}{3 n+2}\right)^{n}\) (c) \(\sum_{n=1}^{\infty}\left(\frac{1}{2}+\frac{1}{n}\right)^{n}\)

Use a CAS to find the first four nonzero terms in the Maclaurin series for each of the following. Check Problems \(43-48\) to see that you get the same answers using the methods of Section \(9.7 .\) \(3 \sin x-2 \exp x\)

Test for convergence or divergence. In some cases, a clever manipulation using the properties of logarithms will simplify the problem. (a) \(\sum_{n=1}^{\infty} \ln \left(1+\frac{1}{n}\right)\) (b) \(\sum_{n=1}^{\infty} \ln \left[\frac{(n+1)^{2}}{n(n+2)}\right]\) (c) \(\sum_{n=2}^{\infty} \frac{1}{(\ln n)^{\ln n}}\) (d) \(\sum_{n=3}^{\infty} \frac{1}{[\ln (\ln n)]^{\ln n}}\) (e) \(\sum_{n=2}^{\infty} \frac{1}{(\ln n)^{4}}\) (f) \(\sum_{n=1}^{\infty}\left[\frac{\ln n}{n}\right]^{2}\)

Show that \(\int_{0}^{\infty}|\sin x| / x d x\) diverges.

Find $$\lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left(\sin \frac{k}{n}\right) \frac{1}{n}$$ Write an equivalent definite integral.

Use a CAS to find the first four nonzero terms in the Maclaurin series for each of the following. Check Problems \(43-48\) to see that you get the same answers using the methods of Section \(9.7 .\) \(\exp \left(x^{2}\right)\)

Determine the order \(n\) of the Maclaurin polynomial for \(4 \tan ^{-1} x\) that is required to approximate \(\pi=4 \tan ^{-1} 1\) to five decimal places, that is, so that \(\left|R_{n}(1)\right| \leq 0.000005\).

Let \(p(n)\) and \(q(n)\) be polynomials in \(n\) with nonnegative coefficients. Give simple conditions that determine the convergence or divergence of \(\sum_{n=1}^{\infty} \frac{p(n)}{q(n)}\).

Show that the graph of \(y=x \sin \frac{\pi}{x}\) on \((0,1]\) has infinite length.

How large must \(N\) be in order for \(S_{N}=\sum_{k=1}^{N}(1 / k)\) just to exceed 4? Note: Computer calculations show that for \(S_{N}\) to exceed \(20, \quad N=272,400,600\), and for \(S_{N}\) to exceed 100 , \(N \approx 1.5 \times 10^{43}\)

Show that $$ \lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left[\frac{1}{1+(k / n)^{2}}\right] \frac{1}{n}=\frac{\pi}{4} $$

Give conditions on \(p\) that determine the convergence or divergence of \(\sum_{n=1}^{\infty} \frac{1}{n^{p}}\left(1+\frac{1}{2^{p}}+\frac{1}{3^{p}}+\cdots+\frac{1}{n^{p}}\right)\).

Find the third-order Maclaurin polynomial for \((1+x)^{1 / 2}\) and bound the error \(R_{3}(x)\) for \(-0.5 \leq x \leq 0.5\).

Prove that if \(\Sigma a_{n}\) diverges and \(\Sigma b_{n}\) converges, then \(\Sigma\left(a_{n}+b_{n}\right)\) diverges.

Using the definition of limit, prove that \(\lim _{n \rightarrow \infty} n /(n+1)\) \(=1 ;\) that is, for a given \(\varepsilon>0\), find \(N\) such that \(n \geq N \Rightarrow|n /(n+1)-1|<\varepsilon\).

Test for convergence or divergence. (a) \(\sum_{n=1}^{\infty} \sin ^{2}\left(\frac{1}{n}\right)\) (b) \(\sum_{n=1}^{\infty} \tan \left(\frac{1}{n}\right)\) (c) \(\sum_{n=1}^{\infty} \sqrt{n}\left[1-\cos \left(\frac{1}{n}\right)\right]\)

Use a CAS to find the first four nonzero terms in the Maclaurin series for each of the following. Check Problems \(43-48\) to see that you get the same answers using the methods of Section \(9.7 .\) \(\exp (\sin x)\)

Find the third-order Maclaurin polynomial for \((1+x)^{3 / 2}\) and bound the error \(R_{3}(x)\) if \(-0.1 \leq x \leq 0\).

Show that it is possible for \(\Sigma a_{n}\) and \(\Sigma b_{n}\) both to diverge and yet for \(\Sigma\left(a_{n}+b_{n}\right)\) to converge.

Use a CAS to find the first four nonzero terms in the Maclaurin series for each of the following. Check Problems \(43-48\) to see that you get the same answers using the methods of Section \(9.7 .\) \((\sin x)(\exp x)\)

Find the third-order Maclaurin polynomial for $$ (1+x)^{-1 / 2} $$ and bound the error \(R_{3}(x)\) if \(-0.05 \leq x \leq 0.05\).

Let \(S=\left\\{x: x\right.\) is rational and \(\left.x^{2}<2\right\\} .\) Convince yourself that \(S\) does not have a least upper bound in the rational numbers, but does have such a bound in the real numbers. In other words, the sequence of rational numbers \(1,1.4,1.41,1.414, \ldots\), has no limit within the rational numbers.

Use a CAS to find the first four nonzero terms in the Maclaurin series for each of the following. Check Problems \(43-48\) to see that you get the same answers using the methods of Section \(9.7 .\) \((\sin x) /(\exp x)\)

Find the fourth-order Maclaurin polynomial for $$ \ln [(1+x) /(1-x)] $$ and bound the error \(R_{4}(x)\) for \(-0.5 \leq x \leq 0.5\).

Let \(r\) be a fixed number with \(|r|<1 .\) Then it can be shown that \(\sum_{k=1}^{\infty} k r^{k}\) converges, say with sum \(S .\) Use the properties of \(\Sigma\) to show that $$ (1-r) S=\sum_{k=1}^{\infty} r^{k} $$ and then obtain a formula for \(S\), thus generalizing Problem 47 a.

Many drugs are eliminated from the body in an exponential manner. Thus, if a drug is given in dosages of size \(C\) at time intervals of length \(t\), the amount \(A_{n}\) of the drug in the body just after the \((n+1)\) st dose is $$ A_{n}=C+C e^{-k t}+C e^{-2 k t}+\cdots+C e^{-n k t} $$ where \(k\) is a positive constant that depends on the type of drug. (a) Derive a formula for \(A\), the amount of drug in the body just after a dose, if a person has been on the drug for a very long time (assume an infinitely long time). (b) Evaluate \(A\) if it is known that one-half of a dose is eliminated from the body in 6 hours and doses of size 2 milligrams are given every 12 hours.