91Ó°ÊÓ

CONCEPT CHECK Alternating Series An alternating series does not meet the first condition of the Alternating Series Test. What can you conclude about the convergence or divergence of the series? Explain.

Graphical Analysis In Exercises 3 and \(4,\) the figures show the graphs of the first 10 terms, and the graphs of the first 10 terms of the sequence of partial sums, of each series. (a) Identify the series in each figure. (b) Which series is a \(p\) -series? Does it converge or diverge? (c) For the series that are not p-series, how do the magnitudes of the terms compare with the magnitudes of the terms of the \(p\) -series? What conclusion can you draw about the convergence or divergence of the series? (d) Explain the relationship between the magnitudes of the terms of the series and the magnitudes of the terms of the partial sums. \(\sum_{n=1}^{\infty} \frac{6}{n^{3 / 2}}, \quad \sum_{n=1}^{\infty} \frac{6}{n^{3 / 2}+3^{\prime}}\) and $$\sum_{n=1}^{\infty} \frac{6}{n^{3 / 2}}, \quad \sum_{n=1}^{\infty} \frac{6}{n^{3 / 2}+3^{\prime}}$$

Using a Power Series In Exercises 19-28, use the power series $$\frac{1}{1+x}=\sum_{n=0}^{\infty}(-1)^{n} x^{n}, \quad|x|<1$$ to find a power series for the function, centered at \(0,\) and determine the interval of convergence. $$f(x)=\ln \left(x^{2}+1\right)$$

Finding the Interval of Convergence In Exercises \(15-38\) , find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) $$\sum_{n=0}^{\infty} \frac{(-1)^{n+1}(x-1)^{n+1}}{n+1}$$

Finding the Interval of Convergence In Exercises \(15-38\) , find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) $$\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2 n}}{n !}$$

Writing an Equivalent Series In Exercises \(45-48,\) write an equivalent series with the index of summation beginning at \(n=1 .\) $$\sum_{n=0}^{\infty}(-1)^{n+1}(n+1) x^{n}$$

Writing an Equivalent Series In Exercises \(45-48,\) write an equivalent series with the index of summation beginning at \(n=1 .\) $$\sum_{n=2}^{\infty} \frac{x^{n-1}}{(7 n-1) !}$$

In Exercises 45-50, find the positive values of p for which the series converges. $$\sum_{n=1}^{\infty} n\left(1+n^{2}\right)^{p}$$

Using the Root Test In Exercises \(39-52,\) use the Root Test to determine the convergence or divergence of the series. $$\sum_{n=1}^{\infty}\left(\frac{\ln n}{n}\right)^{n}$$

Finding a Degree In Exercises 51-56, determine the degree of the Maclaurin polynomial required for the error in the approximation of the function at the indicated value of x to be less than 0.001. $$f(x)=\frac{1}{x+1}, \quad \text { approximate } f(0.2)$$