Chapter 8

\(19-40=\) Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty}(-1)^{n-1} \frac{n}{n^{2}+4} $$

Find the radius of convergence and interval of convergence of the series. $$\sum_{n=1}^{\infty} \frac{n^{2} x^{n}}{2 \cdot 4 \cdot 6 \cdot \cdots \cdot(2 n)}$$

\(9-32\) n Determine whether the sequence converges or diverges. If it converges, find the limit. $$a_{n}=\frac{\tan ^{-1} n}{n}$$

Find a power series representation for \(f,\) and graph \(f\) and several partial sums \(s_{a}(x)\) on the same screen. What happens as \(n\) increases? $$ f(x)=\ln \left(x^{2}+4\right) $$

Find a power series representation for \(f,\) and graph \(f\) and several partial sums \(s_{a}(x)\) on the same screen. What happens as \(n\) increases? $$ f(x)=\ln \left(\frac{1+x}{1-x}\right) $$

A car is moving with speed 20 \(\mathrm{m} / \mathrm{s}\) and acceleration 2\(\mathrm{m} / \mathrm{s}^{2}\) at a given instant. Using a second-degree Taylor polynomial, estimate how far the car moves in the next second. Would it be reasonable to use this polynomial to estimate the distance traveled during the next minute?

Determine whether the series is convergent or divergent. If it is convergent, find its sum. \(\frac{1}{3}+\frac{1}{6}+\frac{1}{9}+\frac{1}{12}+\frac{1}{15}+\cdots\)

Use the binomial series to expand the function as a power serics. State the radius of convergence. $$\sqrt[4]{1-x}$$

\(19-40=\) Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=0}^{\infty} \frac{(-1)^{n}}{5 n+1} $$

If \(\Sigma_{n=0}^{\infty} c_{n} 4^{n}\) is convergent, does it follow that the following series are convergent? $$\text { (a) }\sum_{n=0}^{*} c_{n}(-2)^{n} \quad \text { (b) } \sum_{n=0}^{\infty} c_{n}(-4)^{n}$$

\(9-32\) n Determine whether the sequence converges or diverges. If it converges, find the limit. $$\left\\{n^{2} e^{-n}\right\\}$$

Find a power series representation for \(f,\) and graph \(f\) and several partial sums \(s_{a}(x)\) on the same screen. What happens as \(n\) increases? $$ f(x)=\tan ^{-1}(2 x) $$

The resistivity \(\rho\) of a conducting wire is the reciprocal of the conductivity and is measured in units of ohm-meters \((\Omega-m) .\) The resistivity of a given metal depends on the temperature according to the equation $$\rho(t)=\rho_{20} e^{\alpha(t-20)}$$ where \(t\) is the temperature in \(^{\circ} \mathrm{C} .\) There are tables that list the values of \(\alpha\) (called the temperature coefficient) and \(\rho_{20}\) (the resistivity at \(20^{\circ} \mathrm{C} )\) for various metals. Except at very low temperatures, the resistivity varies almost linearly with temperature and so it is common to approximate the expression for \(\rho(t)\) by its first- or second-degree Taylor polynomial at \(t=20\) . (a) Find expressions for these linear and quadratic approximations. (b) For copper, the tables give \(\alpha=0.0039 /^{\circ} \mathrm{C}\) and \(\rho_{20}=1.7 \times 10^{-8} \Omega-\mathrm{m} .\) Graph the resistivity of copper and the linear and quadratic approximations for \(-250^{\circ} \mathrm{C} \leqslant t \leqslant 1000^{\circ} \mathrm{C}\) (c) For what values of \(t\) does the linear approximation agree with the exponential expression to within one percent?

Determine whether the series is convergent or divergent. If it is convergent, find its sum. \(\frac{1}{3}+\frac{2}{9}+\frac{1}{27}+\frac{2}{81}+\frac{1}{243}+\frac{2}{729}+\cdots\)

Use the binomial series to expand the function as a power serics. State the radius of convergence. $$\sqrt[3]{8+x}$$

Suppose that \(\Sigma_{n=0}^{\infty} C_{n} x^{n}\) converges when \(x=-4\) and diverges when \(x=6 .\) What can be said about the convergence or divergence of the following series? $$\text { (a) } \sum_{n=0}^{\infty} c_{n} \quad \text { (b) } \sum_{n=0}^{\infty} c_{n} 8^{n}$$ $$\text { (c) }\sum_{n=0}^{\infty} c_{n}(-3)^{n} \quad \text { (d) } \sum_{n=0}^{\infty}(-1)^{n} c_{n} 9^{n}$$

\(19-40=\) Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=0}^{\infty} \frac{(-3)^{n}}{(2 n+1) !} $$

\(9-32\) n Determine whether the sequence converges or diverges. If it converges, find the limit. $$a_{n}=\ln (n+1)-\ln n$$

Determine whether the series is convergent or divergent by expressing \(s_{n}\) as a telescoping sum (as in Example 6\() .\) If it is convergent, find its sum. $$\sum_{n=2}^{\infty} \frac{2}{n^{2}-1}$$

Use the binomial series to expand the function as a power serics. State the radius of convergence. $$\frac{1}{(2+x)^{3}}$$

\(19-40=\) Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $$ \sum_{k=1}^{\infty} k\left(\frac{2}{3}\right)^{k} $$

If \(k\) is a positive integer, find the radius of convergence of the series $$\sum_{n=0}^{\infty} \frac{(n !)^{k}}{(k n) !} x^{n}$$

\(9-32\) n Determine whether the sequence converges or diverges. If it converges, find the limit. $$a_{n}=\frac{\cos ^{2} n}{2^{n}}$$

An electric dipole consists of two electric charges of equal magnitude and opposite sign. If the charges are \(q\) and \(-q\) and are located at a distance \(d\) from each other, then the electric field \(E\) at the point \(P\) in the figure is $$E=\frac{q}{D^{2}}-\frac{q}{(D+d)^{2}}$$ By expanding this expression for \(E\) as a series in powers of \(d / D,\) show that \(E\) is approximately proportional to 1\(/ D^{3}\) when \(P\) is far away from the dipole.

Evaluate the indefinite integral as a power series. What is the radius of convergence? $$ \int \frac{t}{1+t^{3}} d t $$

If a water wave with length \(L\) moves with velocity \(v\) across a body of water with depth \(d,\) as in the figure on page \(496,\) then $$v^{2}=\frac{g L}{2 \pi} \tanh \frac{2 \pi d}{L}$$ (a) If the water is deep, show that \(v \approx \sqrt{g L /(2 \pi)}\) (b) If the water is shallow, use the Maclaurin series for tanh to show that \(v \approx \sqrt{g d} .\) (Thus in shallow water the velocity of a wave tends to be independent of the length of the wave.) (c) Use the Alternating Series Estimation Theorem to show that if \(L>10 d\) , then the estimate \(v^{2} \approx g d\) is accurate to within 0.014\(g L .\)

Determine whether the series is convergent or divergent by expressing \(s_{n}\) as a telescoping sum (as in Example 6\() .\) If it is convergent, find its sum. $$\sum_{n=1}^{\infty} \ln \frac{n}{n+1}$$

Let \(p\) and \(q\) be real numbers with \(p

Use the binomial series to expand the function as a power serics. State the radius of convergence. $$(1-x)^{2 / 3}$$

\(19-40=\) Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty} \frac{n !}{100^{n}} $$

\(9-32\) n Determine whether the sequence converges or diverges. If it converges, find the limit. $$a_{n}=2^{-n} \cos n \pi$$

Evaluate the indefinite integral as a power series. What is the radius of convergence? $$ \int x^{2} \ln (1+x) d x $$

If a surveyor measures differences in elevation when making plans for a highway across a desert, corrections must be made for the curvature of the earth. (a) If \(R\) is the radius of the earth and \(L\) is the length of the highway, show that the correction is $$C=R \sec (L / R)-R$$ (b) Use a Taylor polynomial to show that $$C \approx \frac{L^{2}}{2 R}+\frac{5 L^{4}}{24 R^{3}}$$ Compare the corrections given by the formulas in parts (a) and (b) for a highway that is 100 \(\mathrm{km}\) long. (Take the radius of the earth to be 6370 \(\mathrm{km.}\) )

Determine whether the series is convergent or divergent by expressing \(s_{n}\) as a telescoping sum (as in Example 6\() .\) If it is convergent, find its sum. $$\sum_{n=1}^{\infty} \frac{3}{n(n+3)}$$

Is it possible to find a power series whose interval of convergence is \([0, \infty) ?\) Explain.

Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. $$f(x)=\sin \pi x$$

\(19-40=\) Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty} \frac{10^{n}}{(n+1) 4^{2 n+1}} $$

\(9-32\) n Determine whether the sequence converges or diverges. If it converges, find the limit. $$a_{n}=\left(1+\frac{2}{n}\right)^{n}$$

Determine whether the series is convergent or divergent by expressing \(s_{n}\) as a telescoping sum (as in Example 6\() .\) If it is convergent, find its sum. $$\sum_{n=1}^{\infty}\left(e^{1 / n}-e^{1 /(n+1)}\right)$$

Graph the first several partial sums \(s_{n}(x)\) of the series \(\Sigma_{n-0}^{m} x^{n},\) together with the sum function \(f(x)=1 /(1-x)\) on a common screen. On what interval do these partial sums appear to be converging to \(f(x) ?\)

\(19-40=\) Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty} \frac{\sin 4 n}{4^{n}} $$

\(9-32\) n Determine whether the sequence converges or diverges. If it converges, find the limit. $$a_{n}=\frac{\sin 2 n}{1+\sqrt{n}}$$

Evaluate the indefinite integral as a power series. What is the radius of convergence? $$ \int \frac{\tan ^{-1} x}{x} d x $$

Use a power series to approximate the definite integral to six decimal places. $$ \int_{0}^{0.2} \frac{1}{1+x^{5}} d x $$

In Section 4.6 we considered Newton's method for approxi- mating a root \(r\) of the equation \(f(x)=0,\) and from an initial approximation \(x_{1}\) we obtained successive approximations \(x_{2}, x_{3}, \ldots,\) where $$x_{n+1}=x_{n}-\frac{f\left(x_{n}\right)}{f^{\prime}\left(x_{n}\right)}$$ Use Taylor's Formula with \(n=1, a=x_{n},\) and \(x=r\) to show that if \(f^{\prime \prime}(x)\) exists on an interval \(I\) containing \(r, x_{n}\) and \(x_{n+1},\) and \(\left|f^{\prime \prime}(x)\right| \leqslant M,\left|f^{\prime}(x)\right| \geqslant K\) for all \(x \in I,\) then $$ \left|x_{n+1}-r\right| \leqslant \frac{M}{2 K}\left|x_{n}-r\right|^{2}$$ [This means that if \(x_{n}\) is accurate to \(d\) decimal places, then \(x_{n+1}\) is accurate to about 2\(d\) decimal places. More precisely,if the error at stage \(n\) is at most \(10^{-m}\) , then the error at stage \(n+1\) is at most \((M / 2 K) 10^{-2 m} . ]\)

Let \(x=0.99999 \ldots\) (a) Do you think that \(x<1\) or \(x=1 ?\) (b) Sum a geometric series to find the value of \(x\) . (c) How many decimal representations does the number 1 have? (d) Which numbers have more than one decimal representation?

The function \(J_{1}\) defined by $$J_{1}(x)=\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2 n+1}}{n !(n+1) ! 2^{2 n+1}}$$ is called the Bessel function of order \(l\) . (a) Find its domain. (b) Graph the first several partial sums on a common screen. (c) If your CAS has built-in Bessel functions, graph \(J_{1}\) on the same screen as the partial sums in part (b) and observe how the partial sums approximate \(J_{1}\) .

\(19-40=\) Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty} \frac{\cos (n \pi / 3)}{n !} $$

Determine whether the series is convergent or divergent. $$\sum_{n=1}^{\infty} \sin \left(\frac{1}{n}\right)$$

\(9-32\) n Determine whether the sequence converges or diverges. If it converges, find the limit. $$\\{0,1,0,0,1,0,0,0,1, \ldots\\}$$