Chapter 8

Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. $$f(x)=e^{x}+e^{2 x}$$

Use a power series to approximate the definite integral to six decimal places. $$ \int_{0}^{0.4} \ln \left(1+x^{4}\right) d x $$

Use the following outline to prove that \(e\) is an irrational number. (a) If \(e\) were rational, then it would be of the form \(e=p / q\) where \(p\) and \(q\) are positive integers and \(q>2 .\) Use Taylor's Formula to write $$ \begin{array}{c}{\frac{p}{q}=e=1+\frac{1}{1 !}+\frac{1}{2 !}+\ldots+\frac{1}{q !}+\frac{e^{z}}{(q+1) !}} \\ {=s_{q}+\frac{e^{z}}{(q+1) !}}\end{array}$$ where \(0<\)z\(<1\) (b) Show that \(q !\left(e-s_{q}\right)\) is an integer. (c) Show that \(q !\left(e-s_{q}\right)<1\) (d) Use parts (b) and (c) to deduce that \(e\) is irrational.

The function \(A\) defined by $$A(x)=1+\frac{x^{3}}{2 \cdot 3}+\frac{x^{6}}{2 \cdot 3 \cdot 5 \cdot 6}+\frac{x^{9}}{2 \cdot 3 \cdot 5 \cdot 6 \cdot 8 \cdot 9}+\cdots$$ is called an Airy function after the English mathematician and astronomer Sir George Airy \((1801-1892) .\) (a) Find the domain of the Airy function. (b) Graph the first several partial sums on a common screen. (c) If your CAS has built-in Airy functions, graph \(A\) on the same screen as the partial sums in part (b) and observe how the partial sums approximate \(A\) .

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

Determine whether the series is convergent or divergent. $$\sum_{n=1}^{\infty} \frac{n+5}{\sqrt[3]{n^{7}+n^{2}}}$$

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

Use a power series to approximate the definite integral to six decimal places. $$ \int_{0}^{0.1} x \arctan (3 x) d x $$

Express the number as a ratio of integers. \(0 . \overline{8}=0.8888 \ldots\)

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

Find the values of \(p\) for which the series is convergent. $$\sum_{n=2}^{\infty} \frac{1}{n(\ln n)^{p}}$$

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

A function \(f\) is defined by $$f(x)=1+2 x+x^{2}+2 x^{3}+x^{4}+\cdots$$ that is, its coefficients are \(c_{2 n}=1\) and \(c_{2 n+1}=2\) for all \(n \geqslant 0 .\) Find the interval of convergence of the series and find an explicit formula for \(f(x)\)

Use a power series to approximate the definite integral to six decimal places. $$ \int_{0}^{0.3} \frac{x^{2}}{1+x^{4}} d x $$

If \(f(x)=\Sigma_{n-0}^{\infty} c_{n} x^{n},\) where \(c_{n+4}=c_{n}\) for all \(n \geqslant 0,\) find the interval of convergence of the series and a formula for \(f(x)\).

Express the number as a ratio of integers. \(0 . \overline{46}=0.46464646 \ldots\)

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

Find the values of \(p\) for which the series is convergent. $$\sum_{n=1}^{\infty} \frac{\ln n}{n^{p}}$$

\(9-32\) n Determine whether the sequence converges or diverges. If it converges, find the limit. $$a_{n}=\frac{(-3)^{n}}{n !}$$

Show that if \(\lim _{n \rightarrow \infty} \sqrt[n]{\left|c_{n}\right|}=c,\) where \(c \neq 0,\) then the radius of convergence of the power series \(\Sigma c_{n} x^{n}\) is \(R=1 / c\).

Express the number as a ratio of integers. \(2 . \overline{516}=2.516516516 \ldots\)

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

If \(\$ 1000\) is invested at 6\(\%\) interest, compounded annually, then after \(n\) years the investment is worth \(a_{n}=1000(1.06)^{n}\) dollars. (a) Find the first five terms of the sequence \(\left\\{a_{n}\right\\} .\) (b) Is the sequence convergent or divergent? Explain.

Let \(s\) be the sum of a series \(\Sigma a_{n}\) that has been shown to be convergent by the Integral Test and let \(f(x)\) be the function in that test. The remainder after \(n\) terms is $$R_{n}=s-s_{n}=a_{n+1}+a_{n+2}+a_{n+3}+\cdots$$ Thus \(R_{n}\) is the error made when \(s_{n},\) the sum of the first \(n\) terms, is used as an approximation to the total sum s. (a) By comparing areas in a diagram like Figures 3 and 4 (but with \(x \geqslant n ),\) show that $$\int_{n+1}^{\infty} f(x) d x \leqslant R_{n} \leqslant \int_{n}^{\infty} f(x) d x$$ (b) Deduce from part (a) that $$\quad s_{n}+\int_{n+1}^{\infty} f(x) d x \leqslant s \leqslant s_{n}+\int_{n}^{\infty} f(x) d x$$

Show that the function $$ f(x)=\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2 n}}{(2 n) !} $$ is a solution of the differential equation $$ f^{\prime \prime}(x)+f(x)=0 $$

Suppose that the power series \(\Sigma c_{n}(x-a)^{n}\) satisfies \(c_{n} \neq 0\) for all \(n .\) Show that if \(\lim _{n \rightarrow \infty}\left|c_{n} / c_{n+1}\right|\) exists, then it is equal to the radius of convergence of the power series.

Express the number as a ratio of integers. \(10.1 \overline{35}=10.135353535 \ldots\)

\(19-40=\) Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=2}^{\infty}\left(\frac{-2 n}{n+1}\right)^{5 n} $$

Find the first 40 terms of the sequence defined by $$a_{n+1}=\left\\{\begin{array}{ll}{\frac{1}{2} a_{n}} & {\text { if } a_{n} \text { is an even number }} \\ {3 a_{n}+1} & {\text { if } a_{n} \text { is an odd number }}\end{array}\right.$$ and \(a_{1}=11 .\) Do the same if \(a_{1}=25 .\) Make a conjecture about this type of sequence.

Suppose the series \(\Sigma c_{n} x^{n}\) has radius of convergence 2 and the series \(\sum d_{n} x^{n}\) has radius of convergence \(3 .\) What is the radius of convergence of the series \(\Sigma\left(c_{n}+d_{n}\right) x^{n} ?\)

Find the values of \(x\) for which the series converges. Find the sum of the series for those values of \(x .\) $$\sum_{n=1}^{\infty}(-5)^{n} x^{n}$$

Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. $$f(x)=\sin ^{2} x\left[\text { Hint: Use } \sin ^{2} x=\frac{1}{2}(1-\cos 2 x).\right]$$

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

Suppose you know that \(\left\\{a_{n}\right\\}\) is a decreasing sequence and all its terms lie between the numbers 5 and \(8 .\) Explain why the sequence has a limit. What can you say about the value of the limit?

Suppose that the radius of convergence of the power series \(\Sigma c_{n} x^{n}\) is \(R .\) What is the radius of convergence of the power series \(\Sigma c_{n} x^{2 m} ?\)

Find the values of \(x\) for which the series converges. Find the sum of the series for those values of \(x .\) $$\sum_{n=0}^{\infty}(-4)^{n}(x-5)^{n}$$

Find the sum of the series \(\Sigma_{n=1}^{\infty} 1 / n^{5}\) correct to three decimal places.

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

(a) If \(\left\\{a_{n}\right\\}\) is convergent, show that $$\lim _{n \rightarrow \infty} a_{n+1}=\lim _{n \rightarrow \infty} a_{n}$$ (b) A sequence \(\left\\{a_{n}\right\\}\) is defined by \(a_{1}=1\) and \(a_{n+1}=1 /\left(1+a_{n}\right)\) for \(n \geqslant 1 .\) Assuming that \(\left\\{a_{n}\right\\}\) is convergent, find its limit.

(a) Show that the function $$ f(x)=\sum_{n=0}^{\infty} \frac{x^{n}}{n !} $$ is a solution of the differential equation $$ f^{\prime}(x)=f(x) $$ (b) Show that \(f(x)=e^{x}\)

Find the values of \(x\) for which the series converges. Find the sum of the series for those values of \(x .\) $$\sum_{n=0}^{\infty} \frac{(x-2)^{n}}{3^{n}}$$

Find the Maclaurin series of \(f(\) by any method \()\) and its radius of convergence. Graph \(f\) and its first few Taylor polynomials on the same screen. What do you notice about the relationship between these polynomials and \(f ?\) $$f(x)=\cos \left(x^{2}\right)$$

\(19-40=\) Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $$ 1-\frac{1 \cdot 3}{3 !}+\frac{1 \cdot 3 \cdot 5}{5 !}-\frac{1 \cdot 3 \cdot 5 \cdot 7}{7 !}+\cdots $$

\(37-40=\) Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? $$a_{n}=\frac{1}{2 n+3}$$

We have seen that the harmonic series is a divergent series whose terms approach 0. Show that $$\sum_{n=1}^{\infty} \ln \left(1+\frac{1}{n}\right)$$ is another series with this property.

Show that if we want to approximate the sum of the series \(\sum_{n=1}^{\infty} n^{-1.001}\) so that the error is less than 5 in the ninth decimal place, then we need to add more than \(10^{11.301}\) terms!

Find the Maclaurin series of \(f(\) by any method \()\) and its radius of convergence. Graph \(f\) and its first few Taylor polynomials on the same screen. What do you notice about the relationship between these polynomials and \(f ?\) $$f(x)=e^{-x^{2}}+\cos x$$

\(19-40=\) Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $$ \frac{2}{5}+\frac{2 \cdot 6}{5 \cdot 8}+\frac{2 \cdot 6 \cdot 10}{5 \cdot 8 \cdot 11}+\frac{2 \cdot 6 \cdot 10 \cdot 14}{5 \cdot 8 \cdot 11 \cdot 14}+\cdots $$

\(37-40=\) Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? $$a_{n}=\frac{2 n-3}{3 n+4}$$

Let \(f_{n}(x)=(\sin n x) / n^{2}\) . Show that the series \(\Sigma f_{n}(x)\) converges for all values of \(x\) but the series of derivatives \(\Sigma f_{n}^{\prime}(x)\) diverges when \(x=2 n \pi, n\) an integer. For what values of \(x\) does the series \(\Sigma f_{n}^{\prime \prime}(x)\) converge?