Chapter 10

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=1}^{\infty} \frac{n+1}{n^{2}+3 n} \cdot \frac{1}{5 n}\end{equation}

Let \(f(x)\) have derivatives through order \(n\) at \(x=a .\) Show that the Taylor polynomial of order \(n\) and its first \(n\) derivatives have the same values that \(f\) and its first \(n\) derivatives have at \(x=a\) .

Which of the series Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum_{n=1}^{\infty}(-1)^{n} \frac{(2 n) !}{2^{n} n ! n} $$

Determining Convergence or Divergence In Exercises \(17-44,\) use any method to determine if the series converges or diverges. Give reasons for your answer. $$\sum_{n=2}^{\infty} \frac{-n}{(\ln n)^{n}}$$

In Exercises \(37-40,\) find the series' radius of convergence. $$ \sum_{n=1}^{\infty} \frac{(n !)^{2}}{2^{n}(2 n) !} x^{n} $$

In Exercises \(35-40,\) find a formula for the \(n\) th partial sum of the series and use it to determine if the series converges or diverges. If a series converges, find its sum. $$ \sum_{n=1}^{\infty}\left(\cos ^{-1}\left(\frac{1}{n+1}\right)-\cos ^{-1}\left(\frac{1}{n+2}\right)\right) $$

Which of the series in Exercises \(11-40\) converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series' convergence or divergence.) $$ \sum_{n=1}^{\infty} \operatorname{sech} n $$

Use series to evaluate the limits. \begin{equation} \lim _{x \rightarrow 0} \frac{\ln \left(1+x^{3}\right)}{x \cdot \sin x^{2}} \end{equation}

Approximation properties of Taylor polynomials Suppose that \(f(x)\) is differentiable on an interval centered at \(x=a\) and that \(g(x)=b_{0}+b_{1}(x-a)+\cdots+b_{n}(x-a)^{n}\) is a polynomial of degree \(n\) with constant coefficients \(b_{0}, \ldots, b_{n}\) . Let \(E(x)=\) \(f(x)-g(x) .\) Show that if we impose on \(g\) the conditions i) \(E(a)=0\) ii) $$\lim _{x \rightarrow a} \frac{E(x)}{(x-a)^{n}}=0$$ then $$\begin{array}{r}{g(x)=f(a)+f^{\prime}(a)(x-a)+\frac{f^{\prime \prime}(a)}{2 !}(x-a)^{2}+\cdots} \\ {+\frac{f^{(n)}(a)}{n !}(x-a)^{n}}\end{array}$$ Thus, the Taylor polynomial \(P_{n}(x)\) is the only polynomial of degree less than or equal to \(n\) whose error is both zero at \(x=a\) and negligible when compared with \((x-a)^{n}.\)

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=1}^{\infty} \frac{2^{n}+3^{n}}{3^{n}+4^{n}}\end{equation}

Which of the series Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum_{n=1}^{\infty}(-1)^{n} \frac{(n !)^{2} 3^{n}}{(2 n+1) !} $$

In Exercises \(37-40,\) find the series' radius of convergence. $$ \sum_{n=1}^{\infty}\left(\frac{n}{n+1}\right)^{n^{2}} x^{n} $$

Which of the series in Exercises \(11-40\) converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series' convergence or divergence.) $$ \sum_{n=1}^{\infty} \operatorname{sech}^{2} n $$

Determining Convergence or Divergence In Exercises \(17-44,\) use any method to determine if the series converges or diverges. Give reasons for your answer. $$\sum_{n=2}^{\infty} \frac{n}{(\ln n)^{(n / 2)}}$$

In Exercises \(35-40,\) find a formula for the \(n\) th partial sum of the series and use it to determine if the series converges or diverges. If a series converges, find its sum. $$ \sum_{n=1}^{\infty}(\sqrt{n+4}-\sqrt{n+3}) $$

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(27-90\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\left(-\frac{1}{2}\right)^{n} $$

The estimate \(\sqrt{1+x}=1+(x / 2)\) is used when \(x\) is small. Estimate the error when \(|x|<0.01.\)

Quadratic Approximations The Taylor polynomial of order 2 generated by a twice-differentiable function \(f(x)\) at \(x=a\) is called the quadratic approximation of \(f\) at \(x=a.\) find the (a) linearization (Taylor polynomial of order 1) and (b) quadratic approximation of \(f\) at \(x=0\). \(f(x)=\ln (\cos x)\)

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=1}^{\infty} \frac{2^{n}-n}{n 2^{n}}\end{equation}

Which of the series Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum_{n=1}^{\infty}(-1)^{n}(\sqrt{n+1}-\sqrt{n}) $$

Determining Convergence or Divergence In Exercises \(17-44,\) use any method to determine if the series converges or diverges. Give reasons for your answer. $$\sum_{n=1}^{\infty} \frac{n ! \ln n}{n(n+2) !}$$

Find the sum of each series in Exercises \(41-48\) $$ \sum_{n=1}^{\infty} \frac{4}{(4 n-3)(4 n+1)} $$

Quadratic Approximations The Taylor polynomial of order 2 generated by a twice-differentiable function \(f(x)\) at \(x=a\) is called the quadratic approximation of \(f\) at \(x=a.\) find the (a) linearization (Taylor polynomial of order 1) and (b) quadratic approximation of \(f\) at \(x=0\). \(f(x)=e^{\sin x}\)

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=1}^{\infty} \frac{\ln n}{\sqrt{n} e^{n}}\end{equation}

Which of the series Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum_{n=1}^{\infty}(-1)^{n}\left(\sqrt{n^{2}+n}-n\right) $$

Determining Convergence or Divergence In Exercises \(17-44,\) use any method to determine if the series converges or diverges. Give reasons for your answer. $$\sum_{n=1}^{\infty} \frac{(-3)^{n}}{n^{3} 2^{n}}$$

Find the sum of each series in Exercises \(41-48\) $$ \sum_{n=1}^{\infty} \frac{6}{(2 n-1)(2 n+1)} $$

In Exercises \(41-48\) , use Theorem 20 to find the series' interval of convergence and, within this interval, the sum of the series as a function of \(x .\) $$ \sum_{n=0}^{\infty}\left(e^{x}-4\right)^{n} $$

Use the identity \(\sin ^{2} x=(1-\cos 2 x) / 2\) to obtain the Maclaurin series for \(\sin ^{2} x .\) Then differentiate this series to obtain the Maclaurin series for 2 \(\sin x \cos x .\) Check that this is the series for \(\sin 2 x .\)

Quadratic Approximations The Taylor polynomial of order 2 generated by a twice-differentiable function \(f(x)\) at \(x=a\) is called the quadratic approximation of \(f\) at \(x=a.\) find the (a) linearization (Taylor polynomial of order 1) and (b) quadratic approximation of \(f\) at \(x=0\). \(f(x)=1 / \sqrt{1-x^{2}}\)

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\begin{array}{l}{\sum_{n=2}^{\infty} \frac{1}{n !}} \\ {\text { (Hint: First show that }(1 / n !) \leq(1 / n(n-1)) \text { for } n \geq 2 . )}\end{array}\end{equation}

Which of the series Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum_{n=1}^{\infty}(-1)^{n}(\sqrt{n+\sqrt{n}}-\sqrt{n}) $$

Determining Convergence or Divergence In Exercises \(17-44,\) use any method to determine if the series converges or diverges. Give reasons for your answer. $$\sum_{n=1}^{\infty} \frac{(n !)^{2}}{(2 n) !}$$

Find the sum of each series in Exercises \(41-48\) $$ \sum_{n=1}^{\infty} \frac{40 n}{(2 n-1)^{2}(2 n+1)^{2}} $$

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(27-90\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\sin \left(\frac{\pi}{2}+\frac{1}{n}\right) $$

Quadratic Approximations The Taylor polynomial of order 2 generated by a twice-differentiable function \(f(x)\) at \(x=a\) is called the quadratic approximation of \(f\) at \(x=a.\) find the (a) linearization (Taylor polynomial of order 1) and (b) quadratic approximation of \(f\) at \(x=0\). \(f(x)=\cosh x\)

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=1}^{\infty} \frac{(n-1) !}{(n+2) !}\end{equation}

Which of the series Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n}}{\sqrt{n}+\sqrt{n+1}} $$

Determining Convergence or Divergence In Exercises \(17-44,\) use any method to determine if the series converges or diverges. Give reasons for your answer. $$\sum_{n=1}^{\infty} \frac{(2 n+3)\left(2^{n}+3\right)}{3^{n}+2}$$

Are there any values of \(x\) for which \(\sum_{n=1}^{\infty}(1 / n x)\) converges? Give reasons for your answer.

Is it true that if \(\sum_{n=1}^{\infty} a_{n}\) is a divergent series of positive numbers, then there is also a divergent series \(\sum_{n=1}^{\infty} b_{n}\) of positive numbers with \(b_{n}<\) \(a_{n}\) for every \(n ?\) Is there a smallest divergent series of positive numbers? Give reasons for your answers.

Taylor's Theorem and the Mean Value Theorem Explain how the Mean Value Theorem (Section \(4.2,\) Theorem 4 ) is a special case of Taylor's Theorem.

Quadratic Approximations The Taylor polynomial of order 2 generated by a twice-differentiable function \(f(x)\) at \(x=a\) is called the quadratic approximation of \(f\) at \(x=a.\) find the (a) linearization (Taylor polynomial of order 1) and (b) quadratic approximation of \(f\) at \(x=0\). \(f(x)=\sin x\)

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=1}^{\infty} \sin \frac{1}{n}\end{equation}

Which of the series Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum_{n=1}^{\infty}(-1)^{n} \operatorname{sech} n $$

Find the sum of each series in Exercises \(41-48\) $$ \sum_{n=1}^{\infty}\left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\right) $$

Linearizations at inflection points Show that if the graph of a twice- differentiable function \(f(x)\) has an inflection point at \(x=a,\) then the linearization of \(f\) at \(x=a\) is also the quadratic approximation of \(f\) at \(x=a .\) This explains why tangent lines fit so well at inflection points.

Quadratic Approximations The Taylor polynomial of order 2 generated by a twice-differentiable function \(f(x)\) at \(x=a\) is called the quadratic approximation of \(f\) at \(x=a.\) find the (a) linearization (Taylor polynomial of order 1) and (b) quadratic approximation of \(f\) at \(x=0\). \(f(x)=\tan x\)

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=1}^{\infty} \tan \frac{1}{n}\end{equation}

Which of the series Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum_{n=1}^{\infty}(-1)^{n} \operatorname{csch} n $$