Chapter 8

Express the given function as a power series in \(x\) with base point \(0 .\) Calculate the radius of convergence \(R\). \(\frac{x^{3}}{1+x^{4}}\)

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

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. $$ \sum_{n=2}^{\infty}(-1)^{n+1} \frac{n^{2}}{e^{n}} $$

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

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. $$ \sum_{n=1}^{\infty} \frac{2^{n}}{n 3^{n}} $$

State what conclusion, if any, may be drawn from the Divergence Test. $$ \sum_{n=1}^{\infty} \frac{3^{n}+5^{n}}{8^{n}} $$

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

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. $$ \sum_{n=1}^{\infty}(-1)^{n}\left(1+\frac{1}{n}\right) \ln \left(1+\frac{1}{n}\right) $$

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

State what conclusion, if any, may be drawn from the Divergence Test. $$ \sum_{n=1}^{\infty} \frac{5^{n}}{2^{n}+3^{n}} $$

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

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. $$ \sum_{n=2}^{\infty} \cos (\pi n) \sin (\pi / n) $$

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

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

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

Evaluate \(\lim _{n \rightarrow \infty} a_{n}\) for the given sequence \(\left\\{a_{n}\right\\}\). $$ a_{n}=n e^{-2 n} $$

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. $$ \sum_{n=1}^{\infty}(-1)^{n}(\pi / 2-\arctan (n)) $$

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. $$ \sum_{n=1}^{\infty} \frac{\sqrt{n}}{n^{2}+1} $$

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

In each of Exercises \(13-24,\) 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. \(\sum_{n=2}^{\infty}(-1)^{n} \frac{\ln (n)}{n}\)

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

Evaluate \(\lim _{n \rightarrow \infty} a_{n}\) for the given sequence \(\left\\{a_{n}\right\\}\). $$ a_{n}=n^{2} / e^{n} $$

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. $$ \sum_{n=1}^{\infty} \frac{n^{3}}{n^{4.01}+1} $$

State what conclusion, if any, may be drawn from the Divergence Test. $$ \sum_{n=1}^{\infty} \frac{2 n}{3 n^{2}+1} $$

Use formula \((8.7 .2)\) to express the given function as a power series in \(x\) with base point \(0 .\) State the radius of convergence \(R\). \(x \ln \left(1+x^{2}\right)\)

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. \(\sum_{n=2}^{\infty}(-1)^{n} \frac{1}{\ln \left(n^{2}\right)}\)

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

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

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. $$ \sum_{n=1}^{\infty} \frac{2}{2^{n}+1 / 2^{n}} $$

State what conclusion, if any, may be drawn from the Divergence Test. $$ \sum_{n=1}^{\infty} \frac{2 n^{2}}{3 n^{2}+n+1} $$

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. \(\sum_{n=1}^{\infty}(-1)^{n} \frac{\ln (n)}{n^{2}}\)

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

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

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. $$ \sum_{n=1}^{\infty} \frac{1}{n !} $$

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

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. \(\sum_{n=1}^{\infty}(-1)^{n} \cdot \frac{n+1}{n^{3}+1}\)

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

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

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. $$ \sum_{n=1}^{\infty} \frac{1}{n^{n}} $$

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

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. $$ f(x)=x^{2}+e^{1+x} \quad N=2 \quad J=\left[x_{0}, c\right]=[-1.2,-1] $$

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. \(\sum_{n=1}^{\infty}(-1)^{n} \cdot\left(1+\frac{1}{n}\right)\)

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

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

In each of Exercises \(17-28,\) 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. $$ \sum_{n=1}^{\infty} \frac{1}{2 n-1} $$

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

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. \(\sum_{n=1}^{\infty}(-1)^{n} \frac{\sqrt{n}}{n+4}\)

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

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

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. $$ \sum_{n=1}^{\infty} \frac{1}{n+\sqrt{n}} $$