Chapter 9

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

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

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 \(\left|R_{n}(1)\right| \leq 0.000005\).

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 \(20, N=272,400,600\), and for \(S_{N}\) to exceed 100 , \(N \approx 1.5 \times 10^{43}\).

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 \(n \geq N \Rightarrow|n /(n+1)-1|<\varepsilon\).

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

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

Note that $$ \begin{aligned} 1-\frac{1}{2}+& \frac{1}{3}-\frac{1}{4}+\cdots-\frac{1}{2 n} \\ &=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{2 n}-\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}\right) \\ &=\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2 n} \end{aligned} $$

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

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 \(-0.1 \leq x \leq 0\).

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) \(\sum_{n=1}^{\infty} \sqrt{n}\left[1-\cos \left(\frac{1}{n}\right)\right]\)

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.

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

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 \(-0.5 \leq x \leq 0.5\).

Prove that if \(\lim _{n \rightarrow \infty} a_{n}=0\) and \(\left\\{b_{n}\right\\}\) is bounded then \(\lim _{n \rightarrow \infty} a_{n} b_{n}=0 .\)

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.

Find the sum of the series $$ \sum_{k=1}^{\infty} \frac{2^{k}}{\left(2^{k+1}-1\right)\left(2^{k}-1\right)} $$

If \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\}\) both diverge, does it follow that \(\left\\{a_{n}+b_{n}\right\\}\) diverges?

Problem 49 suggests that if \(n\) is odd, then the \(n\)th order Maclaurin polynomial for \(\sin x\) is also the \((n+1)\) st order polynomial, so the error can be calculated using \(R_{n+1}\). Use this result to find how large \(n\) must be so that \(\left|R_{n+1}(x)\right|\) is less than \(0.00005\) for all \(x\) in the interval \(0 \leq x \leq \pi / 2\). (Note, \(n\) must be odd.)

A famous sequence \(\left\\{f_{n}\right\\}\), called the Fibonacci Sequence after Leonardo Fibonacci, who introduced it around A.D. 1200 , is defined by the recursion formula $$ f_{1}=f_{2}=1, \quad f_{n+2}=f_{n+1}+f_{n} $$ (a) Find \(f_{3}\) through \(f_{10}\) - (b) Let \(\phi=\frac{1}{2}(1+\sqrt{5}) \approx 1.618034\). The Greeks called this number the golden ratio, claiming that a rectangle whose dimensions were in this ratio was "perfect." It can be shown that $$ \begin{aligned} f_{n} &=\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n}-\left(\frac{1-\sqrt{5}}{2}\right)^{n}\right] \\\ &=\frac{1}{\sqrt{5}}\left[\phi^{n}-(-1)^{n} \phi^{-n}\right] \end{aligned} $$ Check that this gives the right result for \(n=1\) and \(n=2\). The general result can be proved by induction (it is a nice challenge). More in line with this section, use this explicit formula to prove that \(\lim _{n \rightarrow \infty} f_{n+1} / f_{n}=\phi\). (c) Using the limit just proved, show that \(\phi\) satisfies the equation \(x^{2}-x-1=0\). Then, in another interesting twist, use the Quadratic Formula to show that the two roots of this equation are \(\phi\) and \(-1 / \phi\), two numbers that occur in the explicit formula for \(f_{n}\).

Problem 50 suggests that if \(n\) is even, then the \(n\)th order Maclaurin polynomial for \(\cos x\) is also the \((n+1)\) st order polynomial, so the error can be calculated using \(R_{n+1}\). Use this result to find how large \(n\) must be so that \(\left|R_{n+1}(x)\right|\) is less than \(0.00005\) for all \(x\) in the interval \(0 \leq x \leq \pi / 2\). (Note, \(n\) must be even.)

If an object of rest mass \(m_{0}\) has velocity \(v\), then (according to the theory of relativity) its mass \(m\) is given by \(m=\) \(m_{0} / \sqrt{1-v^{2} / c^{2}}\), where \(c\) is the velocity of light. Explain how physicists get the approximation $$ m \approx m_{0}+\frac{m_{0}}{2}\left(\frac{v}{c}\right)^{2} $$

In Problems 54-59, use the fact that \(\lim _{x \rightarrow \infty} f(x)=\lim _{x \rightarrow 0^{+}} f\left(\frac{1}{x}\right)\) to find the limits. \(\lim _{n \rightarrow \infty}\left(\frac{n-1}{n+1}\right)^{n}\)

In Problems 54-59, use the fact that \(\lim _{x \rightarrow \infty} f(x)=\lim _{x \rightarrow 0^{+}} f\left(\frac{1}{x}\right)\) to find the limits. \(\lim _{n \rightarrow \infty}\left(\frac{2+n^{2}}{3+n^{2}}\right)^{n^{2}}\)

Use Maclaurin's Formula, rather than l'Hôpital's Rule, to find (a) \(\lim _{x \rightarrow 0} \frac{\sin x-x+x^{3} / 6}{x^{5}}\) (b) \(\lim _{x \rightarrow 0} \frac{\cos x-1+x^{2} / 2-x^{4} / 24}{x^{6}}\)

Let \(g(x)=p(x)+x^{n+1} f(x)\), where \(p(x)\) is a polynomial of degree at most \(n\) and \(f\) has derivatives through order \(n\). Show that \(p(x)\) is the Maclaurin polynomial of order \(n\) for \(g\).