Chapter 9

Find the radius of convergence of $$ \sum_{n=1}^{\infty} \frac{1 \cdot 2 \cdot 3 \cdots n}{1 \cdot 3 \cdot 5 \cdots(2 n-1)} x^{2 n+1} $$

By writing \(1 / x=1 /[1-(1-x)]\) and using the known expansion of \(1 /(1-x)\), find the Taylor series for \(1 / x\) in powers of \(x-1\).

Find the power series representation of \(x /\left(x^{2}-3 x+2\right)\). Hint: Use partial fractions.

Find the radius of convergence of $$ \sum_{n=0}^{\infty} \frac{(p n) !}{(n !)^{p}} x^{n} $$ where \(p\) is a positive integer.

Let \(f(x)=(1+x)^{1 / 2}+(1-x)^{1 / 2}\). Find the Maclaurin series for \(f\) and use it to find \(f^{(4)}(0)\) and \(f^{(51)}(0)\).

$$ \left|\frac{4 c}{c+4}\right| ;[0,1] $$

For the series given in Problems \(27-32\), determine how large \(n\) must be so that using the nth partial sum to approximate the series gives an error of no more than \(0.0002 .\) \(\sum_{k=1}^{\infty} \frac{1}{k(k+1)}\)

\(\sum_{n=2}^{\infty}\left(1-\frac{1}{n}\right)^{n}\)

Give an example of two series \(\Sigma a_{n}\) and \(\Sigma b_{n}\), both convergent, such that \(\sum a_{n} b_{n}\) diverges.

Let \(y=y(x)=x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}-\frac{x^{7}}{7 !}+\ldots\). Show that \(y\) satisfies the differential equation \(y^{\prime \prime}+y=0\) with the conditions \(y(0)=0\) and \(y^{\prime}(0)=1\). From this, guess a simple formula for \(y\).

Find the sum \(S(x)\) of \(\sum_{n=0}^{\infty}(x-3)^{n}\). What is the convergence set?

In each case, find the Maclaurin series for \(f(x)\) by use of known series and then use it to calculate \(f^{(4)}(0)\). (a) \(f(x)=e^{x+x^{2}}\) (b) \(f(x)=e^{\sin x}\) (c) \(f(x)=\int_{0}^{x} \frac{e^{t^{2}}-1}{t^{2}} d t\) (d) \(f(x)=e^{\cos x}=e \cdot e^{\cos x-1}\) (e) \(f(x)=\ln \left(\cos ^{2} x\right)\)

For what values of \(p\) does \(\sum_{n=2}^{\infty} 1 /\left[n(\ln n)^{p}\right]\) converge? Explain.

\(\sum_{n=1}^{\infty} \frac{4^{n}+n}{n !}\)

Show that the positive terms of the alternating harmonic series form a divergent series. Show the same for the negative terms.

Let \(\left\\{f_{n}\right\\}\) be the Fibonacci sequence defined by $$ f_{0}=0, \quad f_{1}=1, \quad f_{n+2}=f_{n+1}+f_{n} $$ (See Problem 52 of Section \(9.1\) and Problem 36 of Section 9.6.) If \(F(x)=\sum_{n=0}^{\infty} f_{n} x^{n}\), show that $$ F(x)-x F(x)-x^{2} F(x)=x $$ and then use this fact to obtain a simple formula for \(F(x)\).

Suppose that \(\sum_{n=0}^{\infty} a_{n}(x-3)^{n}\) converges at \(x=-1\). Why can you conclude that it converges at \(x=6 ?\) Can you be sure that it converges at \(x=7\) ? Explain.

One can sometimes find a Maclaurin series by the method of equating coefficients. For example, let $$ \tan x=\frac{\sin x}{\cos x}=a_{0}+a_{1} x+a_{2} x^{2}+\cdots $$ Then multiply by \(\cos x\) and replace \(\sin x\) and \(\cos x\) by their series to obtain $$ \begin{aligned} x-\frac{x^{3}}{6}+\cdots &=\left(a_{0}+a_{1} x+a_{2} x^{2}+\cdots\right)\left(1-\frac{x^{2}}{2}+\cdots\right) \\ &=a_{0}+a_{1} x+\left(a_{2}-\frac{a_{0}}{2}\right) x^{2}+\left(a_{3}-\frac{a_{1}}{2}\right) x^{3}+\cdots \end{aligned} $$ Thus, $$ a_{0}=0, \quad a_{1}=1, \quad a_{2}-\frac{a_{0}}{2}=0, \quad a_{3}-\frac{a_{1}}{2}=-\frac{1}{6}, \quad \ldots $$ so $$ a_{0}=0, \quad a_{1}=1, \quad a_{2}=0, \quad a_{3}=\frac{1}{3}, \quad \ldots $$ and therefore $$ \tan x=0+x+0+\frac{1}{3} x^{3}+\cdots $$ which agrees with Problem 1. Use this method to find the terms through \(x^{4}\) in the series for \(\sec x\).

Does \(\sum_{n=3}^{\infty} 1 /[n \cdot \ln n \cdot \ln (\ln n)]\) converge or diverge? Explain.

\(\sum_{n=1}^{\infty} \frac{n}{2+n 5^{n}}\)

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

$$ \left|\frac{c^{2}+\sin c}{10 \ln c}\right| ;[2,4] $$

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.

$$ \left|\frac{c^{2}-c}{\cos c}\right| ;\left[0, \frac{\pi}{4}\right] $$

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.

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

Explain why a conditionally convergent series can be rearranged to converge to any given number.

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?

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 \(\Sigma a_{n}\) diverges.

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 $$ f(t)= \begin{cases}0 & \text { if } t<0 \\ t^{4} & \text { if } t \geq 0\end{cases} $$ 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 \(\Sigma a_{n}\) diverges.

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

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 .

Let $$ f(x)= \begin{cases}e^{-1 / x^{2}} & \text { if } x \neq 0 \\ 0 & \text { if } x=0\end{cases} $$ (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 \(n \rightarrow \infty)\).

$$ \frac{1}{x-3} ; a=1 $$

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

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

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. $$ \sin x $$

Root Test Prove that if \(a_{n}>0\) and \(\lim _{n \rightarrow \infty}\left(a_{n}\right)^{1 / n}=R\) then \(\Sigma a_{n}\) converges if \(R<1\) and diverges if \(R>1\).

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 \(2\left|a_{k} b_{k}\right| \leq a_{k}^{2}+b_{k}^{2}\).

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

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) \(\sum_{n=1}^{\infty}\left(\frac{1}{2}+\frac{1}{n}\right)^{n}\)

Find $$ \lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left(\sin \frac{k}{n}\right) \frac{1}{n} $$ Hint: 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. $$ 3 \sin x-2 \exp x $$

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

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) \(\sum_{n=1}^{\infty}\left[\frac{\ln n}{n}\right]^{2}\)