Chapter 8

State what conclusion, if any, may be drawn from the Divergence Test. $$ \sum_{n=1}^{\infty} \cos (1 / n) $$

Verify that the Ratio Test yields no information about the convergence of the given series. Use other methods to determine whether the series converges absolutely, converges conditionally, or diverges. \(\sum_{n=1}^{\infty}(-1)^{n} \frac{e^{n}+\ln (n)}{e^{n} \cdot n^{2}}\)

Determine whether the given series converges absolutely, converges conditionally, or diverges. $$ \sum_{n=1}^{\infty}(-1)^{n} \frac{\sqrt{\sqrt{n}+1}}{\sqrt{n^{3}+1}} $$

Evaluate \(\lim _{n \rightarrow \infty} a_{n}\) for the given sequence \(\left\\{a_{n}\right\\}\). $$ a_{n}=\left(\frac{n+1}{n}\right)^{n} $$

State what conclusion, if any, may be drawn from the Divergence Test. $$ \sum_{n=1}^{\infty} \sin (1 / n) $$

In each of Exercises 13-22, a function \(f,\) an order \(N, a\) base point \(c,\) and a point \(x_{0}\) are given. The interval with endpoints \(c\) and \(x_{0}\) is denoted by \(J\) a. Calculate the approximation \(T_{N}\left(x_{0}\right)\) of \(f\left(x_{0}\right)\). b. By showing that \(M=\max _{t \in J}\left|f^{(N+1)}(t)\right|\) is the greater of \(\left|f^{(N+1)}(c)\right|\) and \(\left|f^{(N+1)}\left(x_{0}\right)\right|,\) and by using inequality (8.8.7) of Theorem 2 , find an upper bound for the absolute error \(\left|R_{N}\left(x_{0}\right)\right|=\left|f\left(x_{0}\right)-T_{N}\left(x_{0}\right)\right|\) that results from the approximation of part a. $$ f(x)=\ln \left(1+x^{2}\right) \quad N=2 \quad J=\left[c, x_{0}\right]=[0,0.4] $$

Verify that the Ratio Test yields no information about the convergence of the given series. Use other methods to determine whether the series converges absolutely, converges conditionally, or diverges. \(\sum_{n=1}^{\infty}(-1)^{n} \frac{2^{n}+n}{2^{n}+n^{3}}\)

Determine whether the given series converges absolutely, converges conditionally, or diverges. $$ \sum_{n=1}^{\infty}(-1)^{n+1}(2 / 3)^{n} $$

Evaluate \(\lim _{n \rightarrow \infty} a_{n}\) for the given sequence \(\left\\{a_{n}\right\\}\). $$ a_{n}=\left(\frac{n-1}{n}\right)^{n} $$

Use the Comparison Test for Divergence to show that the given series diverges. State the series that you use for comparison and the reason for its divergence. $$ \sum_{n=1}^{\infty} \frac{1+\ln (n)}{3 n-2} $$

State what conclusion, if any, may be drawn from the Divergence Test. $$ \sum_{n=1}^{\infty}(n /(n+1)-1 /(n+2)) $$

In each of Exercises \(21-26,\) a function \(f\) and a point \(c\) are given. Use the equation $$ \frac{1}{1-(t-c)}=\sum_{n=0}^{\infty}(t-c)^{n}, \quad|t-c|<1 $$ together with some algebra, to express \(f(x)\) as a power series with base point \(c\). State the radius of convergence \(R\). \(f(x)=\frac{1}{6-x} \quad c=5\)

Verify that the Ratio Test yields no information about the convergence of the given series. Use other methods to determine whether the series converges absolutely, converges conditionally, or diverges. \(\sum_{n=1}^{\infty} \frac{(-3)^{n}}{n^{3}+3^{n}}\)

Determine whether the given series converges absolutely, converges conditionally, or diverges. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{(n+\ln (n))}{n^{3 / 2}} $$

In each of Exercises \(21-28,\) a series \(\sum_{n=1}^{\infty} a_{n}\) is given. Calculate the first five partial sums of the series. That is, calculate \(S_{N}=\sum_{n=1}^{N} a_{n}\) for \(N=1,2,3,4,5\). $$ \sum_{n=1}^{\infty} 1 / n $$

Use the Comparison Test for Divergence to show that the given series diverges. State the series that you use for comparison and the reason for its divergence. $$ \sum_{n=1}^{\infty} \frac{2 n+5}{n^{2}+1} $$

A function \(f\) and a point \(c\) are given. Use the equation $$ \frac{1}{1-(t-c)}=\sum_{n=0}^{\infty}(t-c)^{n}, \quad|t-c|<1 $$ together with some algebra, to express \(f(x)\) as a power series with base point \(c\). State the radius of convergence \(R\). \(f(x)=\frac{1}{6-x} \quad c=5\) \(f(x)=\frac{1}{x} \quad c=1\)

Verify that the Ratio Test yields no information about the convergence of the given series. Use other methods to determine whether the series converges absolutely, converges conditionally, or diverges. \(\sum_{n=1}^{\infty}(-1)^{n} \cdot \frac{n !}{n !+2^{n}}\)

Determine whether the given series converges absolutely, converges conditionally, or diverges. $$ \sum_{n=2}^{\infty}(-1)^{n} \frac{n}{\ln ^{3}(n)} $$

A series \(\sum_{n=1}^{\infty} a_{n}\) is given. Calculate the first five partial sums of the series. That is, calculate \(S_{N}=\sum_{n=1}^{N} a_{n}\) for \(N=1,2,3,4,5\). $$ \sum_{n=1}^{\infty} 2^{n} / n ! $$

Use either the word "may" or the word "must" to fill in the blank so that the completed sentence is correct. Explain your answer by referring to a theorem or example. A series that converges __ have summands that tend to \(0 .\)

In each of Exercises 23-34, derive the Maclaurin series of the given function \(f(x)\) by using a known Maclaurin series. $$ f(x)=\sin (2 x) $$

A function \(f\) and a point \(c\) are given. Use the equation $$ \frac{1}{1-(t-c)}=\sum_{n=0}^{\infty}(t-c)^{n}, \quad|t-c|<1 $$ together with some algebra, to express \(f(x)\) as a power series with base point \(c\). State the radius of convergence \(R\). \(f(x)=\frac{1}{6-x} \quad c=5\) \(f(x)=\frac{1}{x+1} \quad c=-3\)

Determine whether the given series converges absolutely, converges conditionally, or diverges. $$ \sum_{n=1}^{\infty} \sin (n) e^{-n} $$

A series \(\sum_{n=1}^{\infty} a_{n}\) is given. Calculate the first five partial sums of the series. That is, calculate \(S_{N}=\sum_{n=1}^{N} a_{n}\) for \(N=1,2,3,4,5\). $$ \sum_{n=1}^{\infty}\left(2^{-n}+1\right) $$

Use the Comparison Test for Divergence to show that the given series diverges. State the series that you use for comparison and the reason for its divergence. $$ \sum_{n=1}^{\infty} \frac{1}{\sqrt{10+n^{2}}} $$

Use either the word "may" or the word "must" to fill in the blank so that the completed sentence is correct. Explain your answer by referring to a theorem or example. If a series diverges, then the Divergence Test __ succeed in proving the divergence.

In each of Exercises 23-34, derive the Maclaurin series of the given function \(f(x)\) by using a known Maclaurin series. $$ f(x)=x \cos (x) $$

A function \(f\) and a point \(c\) are given. Use the equation $$ \frac{1}{1-(t-c)}=\sum_{n=0}^{\infty}(t-c)^{n}, \quad|t-c|<1 $$ together with some algebra, to express \(f(x)\) as a power series with base point \(c\). State the radius of convergence \(R\). \(f(x)=\frac{1}{6-x} \quad c=5\) \(f(x)=\frac{1}{x-4} \quad c=1\)

Verify that the Ratio Test yields no information about the convergence of the given series. Use other methods to determine whether the series converges absolutely, converges conditionally, or diverges. \(\sum_{n=1}^{\infty}(-1)^{n} \sin \left(\frac{1}{n}\right)\)

Determine whether the given series converges absolutely, converges conditionally, or diverges. $$ \sum_{n=1}^{\infty}(-1)^{n} \frac{\ln ^{3}(n)}{n^{1 / 3}} $$

A series \(\sum_{n=1}^{\infty} a_{n}\) is given. Calculate the first five partial sums of the series. That is, calculate \(S_{N}=\sum_{n=1}^{N} a_{n}\) for \(N=1,2,3,4,5\). $$ \sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n+1}\right) $$

Use the Comparison Test for Divergence to show that the given series diverges. State the series that you use for comparison and the reason for its divergence. $$ \sum_{n=1}^{\infty} \frac{2+\sin (n)}{\sqrt{n}} $$

Use either the word "may" or the word "must" to fill in the blank so that the completed sentence is correct. Explain your answer by referring to a theorem or example. If the partial sums of an infinite series are bounded, then the series __ converge.

In each of Exercises 23-34, derive the Maclaurin series of the given function \(f(x)\) by using a known Maclaurin series. $$ f(x)=x^{2} \sin (x / 2) $$

A function \(f\) and a point \(c\) are given. Use the equation $$ \frac{1}{1-(t-c)}=\sum_{n=0}^{\infty}(t-c)^{n}, \quad|t-c|<1 $$ together with some algebra, to express \(f(x)\) as a power series with base point \(c\). State the radius of convergence \(R\). \(f(x)=\frac{1}{6-x} \quad c=5\) \(f(x)=\frac{1}{2 x+5} \quad c=-1\)

In each of Exercises \(25-34,\) use the Root Test to determine the convergence or divergence of the given series. \(\sum_{n=1}^{\infty} n^{-n / 2}\)

Determine whether the given series converges absolutely, converges conditionally, or diverges. $$ \sum_{n=1}^{\infty}(-1)^{n}\left(\frac{\ln (n)}{n}\right)^{3} $$

A series \(\sum_{n=1}^{\infty} a_{n}\) is given. Calculate the first five partial sums of the series. That is, calculate \(S_{N}=\sum_{n=1}^{N} a_{n}\) for \(N=1,2,3,4,5\). $$ \sum_{n=1}^{\infty} 2^{n} / 3^{n} $$

Use the Comparison Test for Divergence to show that the given series diverges. State the series that you use for comparison and the reason for its divergence. $$ \sum_{n=1}^{\infty} \frac{3^{n}+1}{3^{n}+2^{n}} $$

Use either the word "may" or the word "must" to fill in the blank so that the completed sentence is correct. Explain your answer by referring to a theorem or example. If a series diverges, then its terms __ diverge.

In each of Exercises 23-34, derive the Maclaurin series of the given function \(f(x)\) by using a known Maclaurin series. $$ f(x)=x^{3}+\cos \left(x^{2}\right) $$

Use the Root Test to determine the convergence or divergence of the given series. \(\sum_{n=1}^{\infty} \frac{2^{3 n}}{3^{2 n}}\)

Determine whether the given series converges absolutely, converges conditionally, or diverges. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n}}{1+\sin (n)} $$

A series \(\sum_{n=1}^{\infty} a_{n}\) is given. Calculate the first five partial sums of the series. That is, calculate \(S_{N}=\sum_{n=1}^{N} a_{n}\) for \(N=1,2,3,4,5\). $$ \sum_{n=1}^{\infty} 1 / n^{2} $$

Use the Comparison Test for Divergence to show that the given series diverges. State the series that you use for comparison and the reason for its divergence. $$ \sum_{n=1}^{\infty} \frac{2^{n}}{n^{2}} $$

Use either the word "may" or the word "must" to fill in the blank so that the completed sentence is correct. Explain your answer by referring to a theorem or example. If the partial sums of an infinite series diverge, then the series __ diverge.

In each of Exercises 23-34, derive the Maclaurin series of the given function \(f(x)\) by using a known Maclaurin series. $$ f(x)=5 \cos (2 x)-4 \sin (3 x) $$

In Exercises \(27-38\), compute the Taylor polynomial \(T_{N}\) of the given function \(f\) with the given base point \(c\) and given order \(N\). \(f(x)=\cos (x)\) \(N=4 \quad c=\pi / 3\)

Use the Root Test to determine the convergence or divergence of the given series. \(\sum_{n=1}^{\infty} \frac{n}{2^{n}}\)