Chapter 30

\(f(x)=12+3(x-1)+5(x-1)^{2}+7(x-1)^{3} .\) Find the following. (a) \(f^{\prime}(1)\) (b) \(f^{\prime \prime}(1)\) (c) \(f^{\prime \prime \prime}(1)\) (d) \(f(1)\)

Determine whether the series converges or diverges. In this set of problems knowledge of the Limit Comparison Test is assumed. \(\sum_{k=1}^{\infty} \frac{1}{e^{k}-1}\)

Use any method to find the Maclaurin series for \(f(x) .\) (Strive for efficiency.) Determine the radius of convergence. $$ f(x)=\sin 3 x $$

\(f(x)=\sqrt{3}+12(x-5)^{3}+17(x-5)^{6} .\) Find the following. (a) \(f(5)\) (b) \(f^{\prime \prime}(5)\) (c) \(f^{\prime \prime \prime}(5)\) (d) \(f^{(6)}(5)\)

Determine whether the series converges or diverges. In this set of problems knowledge of the Limit Comparison Test is assumed. \(\sum_{n=2}^{\infty} \frac{n-1}{2 n^{2}-n}\)

Use any method to find the Maclaurin series for \(f(x) .\) (Strive for efficiency.) Determine the radius of convergence. $$ f(x)=\cos \left(\frac{x}{2}\right) $$

Compute the sixth degree Taylor polynomial generated by \(\sin x\) about \(x=\pi\).

Determine whether the series converges or diverges. In this set of problems knowledge of the Limit Comparison Test is assumed. \(\sum_{k=1}^{\infty} \frac{2 k^{2}-k}{3 k^{4}+1}\)

Use any method to find the Maclaurin series for \(f(x) .\) (Strive for efficiency.) Determine the radius of convergence. $$ f(x)=3 e^{2 x} $$

Compute the sixth degree Taylor polynomial generated by \(\cos x\) about \(x=-\frac{\pi}{2}\).

Determine whether the series converges or diverges. In this set of problems knowledge of the Limit Comparison Test is assumed. \(\sum_{k=3}^{\infty} \frac{k}{2 k^{3}-2}\)

Use any method to find the Maclaurin series for \(f(x) .\) (Strive for efficiency.) Determine the radius of convergence. $$ f(x)=\cos \left(x^{2}\right) $$

Let \(f(x)=(1+x)^{p}\), where \(p\) is a constant, \(p \neq 0,1,2,3,4,5\). (a) Compute the third degree Taylor polynomial for \(f(x)\) around \(x=0\). (b) Compute the fth degree Taylor polynomial for \(f(x)\) around \(x=0\).

Determine whether the series converges or diverges. In this set of problems knowledge of the Limit Comparison Test is assumed. \(\sum_{n=2}^{\infty} \frac{1}{\sqrt{n^{2}-n}}\)

Use any method to find the Maclaurin series for \(f(x) .\) (Strive for efficiency.) Determine the radius of convergence. $$ f(x)=3^{x} $$

Determine whether the series converges or diverges. In this set of problems knowledge of the Limit Comparison Test is assumed. \(\sum_{n=2}^{\infty} \frac{n+1}{\ln n}\)

Use any method to find the Maclaurin series for \(f(x) .\) (Strive for efficiency.) Determine the radius of convergence. $$ f(x)=x^{2} \cos (-x) $$

Determine whether the series converges or diverges. In this set of problems knowledge of the Limit Comparison Test is assumed. \(\sum_{k=2}^{\infty} \frac{2^{k}}{5^{k}-5}\)

Use any method to find the Maclaurin series for \(f(x) .\) (Strive for efficiency.) Determine the radius of convergence. $$ f(x)=\cos ^{2} x $$

(a) Let \(\sum_{k=1}^{\infty} a_{k}\) be a convergent series with \(0

Use any method to find the Maclaurin series for \(f(x) .\) (Strive for efficiency.) Determine the radius of convergence. \(f(x)=(a+x)^{p}\), where " \(a\) " and " \(p\) " are constants and \(p\) is not a positive integer.

In Problems 34 through 41, determine whether the series converges or diverges. In this set of problems knowledge of all the convergence tests from the chapter is assumed. \(\sum_{k=1}^{\infty} \frac{3 k}{k^{1}}\)

Use any method to find the Maclaurin series for \(f(x) .\) (Strive for efficiency.) Determine the radius of convergence. $$f(x)=\frac{1}{2 x+3}$$

Determine whether the series converges or diverges. In this set of problems knowledge of all the convergence tests from the chapter is assumed. \(\sum_{n=1}^{\infty} \frac{n^{3}}{2^{n}}\)

Pathological Example: Let \(f(x)=\left\\{\begin{array}{ll}e^{-\frac{1}{x^{2}}} & \text { for } x \neq 0 \\ 0 & \text { for } x=0\end{array}\right.\) (a) Graph \(f(x)\) on the following domains: \([-20,20],[-2,2]\), and \([-0.5,0.5]\). (A graphing instrument can be used.) (b) It can be shown that \(f\) is infinitely differentiable at \(x=0\) and that \(f^{(k)}(0)=0\) for all \(k\). Conclude that the Maclaurin series for \(f(x)\) converges for all \(x\) but only converges to \(f(x)\) at \(x=0\).

Determine whether the series converges or diverges. In this set of problems knowledge of all the convergence tests from the chapter is assumed. \(\sum_{n=1}^{\infty} \frac{3^{n}}{2^{n}}\)

Find the Maclaurin series for \(\frac{1}{\sqrt{e^{x}}}\). What is its radius of convergence?

Determine whether the series converges or diverges. In this set of problems knowledge of all the convergence tests from the chapter is assumed. \(\sum_{k=1}^{\infty} \frac{k^{3}}{k !}\)

For \(x \in(-1,1], \ln (1+x)=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+\cdots+(-1)^{k} \frac{x^{k+1}}{k+1}+\cdots\) (a) Find the Maclaurin series for \(\ln (1+2 x)\). What is its interval of convergence? (b) Find the Maclaurin series for \(\ln (e+e x)\). What is its interval of convergence? (c) Find the Maclaurin series for \(\log _{10}(1+x)\).

Determine whether the series converges or diverges. In this set of problems knowledge of all the convergence tests from the chapter is assumed. \(\sum_{k=1}^{\infty} \frac{k !}{k^{3} 3^{k}}\)

Discover something wonderful. We know \(e^{u}=1+u+\frac{u^{2}}{2 !}+\frac{u^{3}}{3 !}+\cdots+\frac{u^{k}}{k !}+\cdots\) for all real \(u\). Now define \(e\) raised to a complex number, \(a+b i\) where \(i=\sqrt{-1}\), to be \(e^{a} \cdot e^{b i}\) where \(e^{b i}=1+(b i)+\frac{(b i)^{2}}{2 !}+\frac{(b i)^{3}}{3 !}+\cdots+\frac{(b i)^{k}}{k !}+\cdots\) (a) Use the fact that \(i^{2}=-1, i^{3}=-i\), and \(i^{4}=1\) to simplify the expression for \(e^{i b}\) Gather together the real terms (the ones without \(i\) 's) and the terms with a factor of i. Express \(e^{i b}\) as a sum of two familiar functions (one of them multiplied by \(i\) ). (b) Use your answer to part (a) to evaluate \(e^{\pi i}\).

Determine whether the series converges or diverges. In this set of problems knowledge of all the convergence tests from the chapter is assumed. \(\sum_{k=2}^{\infty} \frac{2}{(\ln k)^{k}}\)

The hyperbolic functions, hyperbolic cosine, abbreviated cosh, and hyperbolic sine, abbreviated sinh, are defined as follows. $$\cosh x=\frac{e^{x}+e^{-x}}{2} \quad \sinh x=\frac{e^{x}-e^{-x}}{2}$$ (a) Graph \(\cosh x\) and \(\sinh x\), each on its own set of axes. Do this without using a computer or graphing calculator, except possibly to check your work. (b) Find the Maclaurin series for \(\cosh x\). (c) Find the MacLaurin series for \(\sinh x\). Remark: From the graphs of \(\cosh x\) and \(\sinh x\) one might be surprised by the choice of names for these functions. After finding their Maclaurin series the choice should seem more natural. (d) Do some research and find out how these functions, known as hyperbolic functions, are used. The arch in St. Louis, the shape of many pottery kilns, and the shape of a hanging cable are all connected to hyperbolic trigonometric functions.

Determine whether the series converges or diverges. In this set of problems knowledge of all the convergence tests from the chapter is assumed. \(\sum_{k=1}^{\infty}\left(1+\frac{1}{k}\right)^{k}\)

Determine whether the series converges or diverges. In this set of problems knowledge of all the convergence tests from the chapter is assumed. \(\sum_{k=1}^{\infty}\left(\frac{k^{2}-3 k}{5 k^{2+1}}\right)^{k}\)

For what values of \(n, n\) a positive integer, does \(\sum_{k=1}^{\infty} \frac{k^{n}}{k !}\) converge?

For what values of \(r\) does \(\sum_{k=1}^{\infty} \frac{r^{k}}{k !}\) converge?

Determine whether the series converges absolutely, converges conditionally, or diverges. Explain your reasoning carefully. \(\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k \sqrt{2 k}}\)

Determine whether the series converges absolutely, converges conditionally, or diverges. Explain your reasoning carefully. \(\sum_{k=1}^{\infty} \frac{\cos k}{k^{3}}\)

Determine whether the series converges absolutely, converges conditionally, or diverges. Explain your reasoning carefully. \(\sum_{k=1}^{\infty} \frac{\sin (2 k)}{2^{k}}\)

Determine whether the series converges absolutely, converges conditionally, or diverges. Explain your reasoning carefully. \(\sum_{n=3}^{\infty} \frac{(-1)^{n}}{10 \sqrt{n}}\)

Determine whether the series converges absolutely, converges conditionally, or diverges. Explain your reasoning carefully. \(\sum_{k=1}^{\infty} \frac{(-1)^{k} 5^{k}}{k !}\)

Determine whether the series converges absolutely, converges conditionally, or diverges. Explain your reasoning carefully. \(\sum_{k=1}^{\infty} \frac{(-k)^{k}}{k !}\)

Determine whether the series converges absolutely, converges conditionally, or diverges. Explain your reasoning carefully. \(\sum_{k=2}^{\infty} \frac{(-1)^{k}}{3 k^{3}+3}\)

Does the series \(\sum_{k=2}^{\infty} \frac{(-1)^{k} \ln k}{k}\) converge absolutely, converge conditionally, or diverge? Explain your reasoning carefully and justify your assertions.

Prove the following version of the Integral Test. (It s a slightly weaker version than the one stated in this section.) Let \(\sum_{k=1}^{\infty} a_{k}\) be a series such that \(a_{k}=f(k)\) for \(k=1,2,3, \ldots\) where the function \(f\) is positive, continuous, and decreasing on \([1, \infty)\). (a) If \(\int_{1}^{\infty} f(x) d x\) converges, then \(\sum_{k=1}^{\infty} a_{k}\) converges. (b) If \(\int_{1}^{\infty} f(x) d x\) diverges, then \(\sum_{k=1}^{\infty} a_{k}\) diverges.

In Problems 54 through 59, use the Ratio Test or Root Test to find the radius of convergence of the power series given. \(\sum_{k=1}^{\infty}(-1)^{k} \frac{(2 x)^{k}}{k !}\)

Use the Ratio Test or Root Test to find the radius of convergence of the power series given. \(\sum_{k=1}^{\infty}(-1)^{k} \frac{(3 x)^{k}}{k}\)

Use the Ratio Test or Root Test to find the radius of convergence of the power series given. \(\sum_{k=1}^{\infty} k\left(\frac{x}{2}\right)^{k}\)

Use the Ratio Test or Root Test to find the radius of convergence of the power series given. \(\sum_{n=1}^{\infty} \frac{n(x-5)^{n}}{(2 n) !}\)