Chapter 8

In each of Exercises \(1-10\), express the given function as a power series in \(x\) with base point \(0 .\) Calculate the radius of convergence \(R\). \(\frac{1}{1-2 x}\)

In each of Exercises \(1-12,\) use the Ratio Test to determine the convergence or divergence of the given series. \(\sum_{n=1}^{\infty} \frac{n}{e^{n}}\)

Use Theorem 2 and, where necessary, limit formula (8.5.1) to calculate the radius of convergence \(R\). Determine the interval of convergence \(I\) by checking the endpoints. $$ \sum_{n=0}^{\infty}(2 x)^{n} $$

The given series may be shown to converge by using the Alternating Series Test. Show that the hypotheses of the Alternating Series Test are satisfied. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^{3}+1} $$

In each of Exercises \(1-16,\) use the Comparison Test for Convergence to show that the given series converges. State the series that you use for comparison and the reason for its convergence. $$ \sum_{n=1}^{\infty} \frac{n}{n^{3}+1} $$

In each of Exercises \(1-20,\) evaluate \(\lim _{n \rightarrow \infty} a_{n}\) for the given sequence \(\left\\{a_{n}\right\\}\). $$ a_{n}=\frac{2 n^{2}-1+n^{3}}{5 n^{3}+n+2} $$

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

Express the given function as a power series in \(x\) with base point \(0 .\) Calculate the radius of convergence \(R\). \(\frac{x}{1+x}\)

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

Use Theorem 2 and, where necessary, limit formula (8.5.1) to calculate the radius of convergence \(R\). Determine the interval of convergence \(I\) by checking the endpoints. $$ \sum_{n=0}^{\infty}\left(\frac{-x}{3}\right)^{n} $$

The given series may be shown to converge by using the Alternating Series Test. Show that the hypotheses of the Alternating Series Test are satisfied. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n}}{\sqrt{n}} $$

Use the Comparison Test for Convergence to show that the given series converges. State the series that you use for comparison and the reason for its convergence. $$ \sum_{n=1}^{\infty} \frac{5 n-4}{n^{9 / 4}} $$

Evaluate \(\lim _{n \rightarrow \infty} a_{n}\) for the given sequence \(\left\\{a_{n}\right\\}\). $$ a_{n}=\frac{3 n^{2}+n+4}{2 n^{3}+1} $$

State what conclusion, if any, may be drawn from the Divergence Test. $$ \sum_{n=1}^{\infty} \frac{1}{\sqrt{1+\sqrt{n}}} $$

Express the given function as a power series in \(x\) with base point \(0 .\) Calculate the radius of convergence \(R\). \(\frac{1}{2-x}\)

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

The given series may be shown to converge by using the Alternating Series Test. Show that the hypotheses of the Alternating Series Test are satisfied. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n}}{\sqrt{n}} $$

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

Evaluate \(\lim _{n \rightarrow \infty} a_{n}\) for the given sequence \(\left\\{a_{n}\right\\}\). $$ a_{n}=\frac{2^{n+1}+5}{2^{n}+3} $$

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

Express the given function as a power series in \(x\) with base point \(0 .\) Calculate the radius of convergence \(R\). \(\frac{1}{1+x^{2}}\)

Use the Ratio Test to determine the convergence or divergence of the given series. \(\sum_{n=1}^{\infty} \frac{10^{n}}{n !}\)

The given series may be shown to converge by using the Alternating Series Test. Show that the hypotheses of the Alternating Series Test are satisfied. $$ \sum_{n=2}^{\infty} \frac{(-1)^{n}}{\ln (n)} $$

Use the Comparison Test for Convergence to show that the given series converges. State the series that you use for comparison and the reason for its convergence. $$ \sum_{n=1}^{\infty} \frac{1}{(\sqrt{n}+\sqrt{2})^{4}} $$

Evaluate \(\lim _{n \rightarrow \infty} a_{n}\) for the given sequence \(\left\\{a_{n}\right\\}\). $$ a_{n}=\frac{3^{2 n}+2}{9^{n+1}+1} $$

State what conclusion, if any, may be drawn from the Divergence Test. $$ \sum_{n=1}^{\infty} \frac{3^{n}}{3^{n}+4} $$

Express the given function as a power series in \(x\) with base point \(0 .\) Calculate the radius of convergence \(R\). \(\frac{1}{4+x}\)

Use the Ratio Test to determine the convergence or divergence of the given series. \(\sum_{n=1}^{\infty} \frac{n^{100}}{n !}\)

The given series may be shown to converge by using the Alternating Series Test. Show that the hypotheses of the Alternating Series Test are satisfied. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{(2 / 3)^{n}} $$

Use the Comparison Test for Convergence to show that the given series converges. State the series that you use for comparison and the reason for its convergence. $$ \sum_{n=1}^{\infty} \frac{2+\sin (n)}{n^{4}} $$

Evaluate \(\lim _{n \rightarrow \infty} a_{n}\) for the given sequence \(\left\\{a_{n}\right\\}\). $$ a_{n}=\frac{3^{n}+2 \cdot 5^{n}}{2^{n}+3 \cdot 5^{n}} $$

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

Express the given function as a power series in \(x\) with base point \(0 .\) Calculate the radius of convergence \(R\). \(\frac{1}{9-x^{2}}\)

Use the Ratio Test to determine the convergence or divergence of the given series. \(\sum_{n=1}^{\infty} \frac{n !}{11^{n}(n+12)^{13}}\)

The given series may be shown to converge by using the Alternating Series Test. Show that the hypotheses of the Alternating Series Test are satisfied. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n !} $$

Evaluate \(\lim _{n \rightarrow \infty} a_{n}\) for the given sequence \(\left\\{a_{n}\right\\}\). $$ a_{n}=\frac{2^{n}+5^{n}}{7^{n}} $$

Use the Comparison Test for Convergence to show that the given series converges. State the series that you use for comparison and the reason for its convergence. $$ \sum_{n=1}^{\infty} \frac{2 n-1}{n e^{n}} $$

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

Express the given function as a power series in \(x\) with base point \(0 .\) Calculate the radius of convergence \(R\). \(\frac{x^{2}+1}{1-x^{2}}\)

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

The given series may be shown to converge by using the Alternating Series Test. Show that the hypotheses of the Alternating Series Test are satisfied. $$ \sum_{n=1}^{\infty} \frac{(-4 / 5)^{n}}{n+2} $$

Evaluate \(\lim _{n \rightarrow \infty} a_{n}\) for the given sequence \(\left\\{a_{n}\right\\}\). $$ a_{n}=\frac{6 n+\cos (n)}{3 n+2-\sin \left(n^{2}\right)} $$

Use the Comparison Test for Convergence to show that the given series converges. State the series that you use for comparison and the reason for its convergence. $$ \sum_{n=1}^{\infty} \frac{n+2}{2 n^{5 / 2}+3} $$

State what conclusion, if any, may be drawn from the Divergence Test. $$ \sum_{n=1}^{\infty} \frac{3^{n}}{4^{n}+3} $$

Express the given function as a power series in \(x\) with base point \(0 .\) Calculate the radius of convergence \(R\). \(\frac{x}{1+x^{2}}\)

Use the Ratio Test to determine the convergence or divergence of the given series. \(\sum_{n=1}^{\infty} \frac{\sqrt{n !}}{n^{5} \cdot 7^{n}}\)

The given series may be shown to converge by using the Alternating Series Test. Show that the hypotheses of the Alternating Series Test are satisfied. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n}{n^{2}+1} $$

Evaluate \(\lim _{n \rightarrow \infty} a_{n}\) for the given sequence \(\left\\{a_{n}\right\\}\). $$ a_{n}=\frac{6 n+\cos (n)}{3 n+2-\sin \left(n^{2}\right)} $$

Use the Comparison Test for Convergence to show that the given series converges. State the series that you use for comparison and the reason for its convergence. $$ \sum_{n=1}^{\infty} \frac{n^{2}+2 n+10}{2 n^{4}} $$

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