Chapter 11

Find the Maclaurin series for \( f(x) \) using the definition of a Maclaurin series. [ Assume that \( f \) has a power series expansion. Do not show that \( R_n (x) \to 0. \)] Also find the associated radius of convergence. \( f(x) = x \cos x \)

Find a power series representation for the function and determine the radius of convergence. \( f(x) = x^2 \tan^{-1} (x^3) \)

Find the radius of convergence and interval of convergence of the series. \( \sum_{n = 1}^{\infty} \frac {( - 1)^n}{(2n - 1)2^n} (x - 1)^n \)

Test the series for convergence or divergence. \( \displaystyle \sum_{n = 1}^{\infty} \frac {\sqrt{n^4 + 1}}{n^3 + n} \)

Use the Ratio Test to determine whether the series is convergent or divergent. \( \displaystyle \sum_{n = 1}^{\infty} \frac {n^{10}}{( - 10)^{n+1}} \)

Determine whether the series converges or diverges. \( \displaystyle \sum_{n = 1}^{\infty} \frac {1}{n^n} \)

Determine whether the series is convergent or divergent. \( \displaystyle \sum_{n = 1}^{\infty} \frac {\sqrt n}{1 + n^{3/2}} \)

(a) Explain the difference between \( \displaystyle \sum_{i = 1}^{n} a_i \) and \( \displaystyle \sum_{j = 1}^{n} a_j \) (b) Explain the difference between \( \displaystyle \sum_{i = 1}^{n} a_i \) and \( \displaystyle \sum_{i = 1}^{n} a_j \)

Find a formula for the general term \( a_n \) of the sequence, assuming that the pattern of the first few terms continues. \( \left\\{\begin{array} 5, 8, 11, 14, 17, . . . . .\end{array}\right\\} \)

(a) Approximate \( f \) by a Taylor polynomial with degree \( n \) at the number \( a. \) (b) Use Taylor's Inequality to estimate the accuracy of the approximation \( f(x) \approx T_n(x) \) when \( x \) lies in the given interval. (c) Check you result in part (b) by graphing \( \mid R_n(x) \mid . \) \( f (x) = \sec x, \) \( a = 0, \) \( n = 2, \) \( 0.2 \le x \le 0.2 \)

Find the Maclaurin series for \( f(x) \) using the definition of a Maclaurin series. [ Assume that \( f \) has a power series expansion. Do not show that \( R_n (x) \to 0. \)] Also find the associated radius of convergence. \( f(x) = \sinh x \)

Find a power series representation for the function and determine the radius of convergence. \( f(x) = \frac {x}{(1 + 4x)^2} \)

Find the radius of convergence and interval of convergence of the series. \( \sum_{n = 2}^{\infty} \frac {(x + 2)^n}{2^n \ln n} \)

Use the Ratio Test to determine whether the series is convergent or divergent. \( \displaystyle \sum_{n = 1}^{\infty} \frac {\cos (n \pi /3)}{n!} \)

Test the series for convergence or divergence. $$ \sum_{n=1}^{\infty}(-1)^{n} \sin \left(\frac{\pi}{n}\right) $$

Determine whether the series converges or diverges. \( \displaystyle \sum_{n = 1}^{\infty} \frac {1}{\sqrt {n^2 + 1}} \)

Determine whether the series is convergent or divergent. \( \displaystyle \sum_{n = 1}^{\infty} \frac {1}{n^2 + 4} \)

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. \( 3 - 4 + \frac {16}{3} - {64}{9} + \cdot \cdot \cdot \)

Find a formula for the general term \( a_n \) of the sequence, assuming that the pattern of the first few terms continues. \( \left\\{\begin{array} \frac {1}{2}, - \frac {4}{3}, \frac {9}{4}, - \frac {16}{5}, \frac {25}{6}, . . . . .\end{array}\right\\} \)

(a) Approximate \( f \) by a Taylor polynomial with degree \( n \) at the number \( a. \) (b) Use Taylor's Inequality to estimate the accuracy of the approximation \( f(x) \approx T_n(x) \) when \( x \) lies in the given interval. (c) Check you result in part (b) by graphing \( \mid R_n(x) \mid . \) \( f (x) = \ln(1 + 2x), \) \( a = 1, \) \( n = 3, \) \( 0.5 \le x \le 1.5 \)

Find the Maclaurin series for \( f(x) \) using the definition of a Maclaurin series. [ Assume that \( f \) has a power series expansion. Do not show that \( R_n (x) \to 0. \)] Also find the associated radius of convergence. \( f(x) = \cosh x \)

Find a power series representation for the function and determine the radius of convergence. \( f(x) = \left( \frac {x}{2 - x} \right)^3 \)

Find the radius of convergence and interval of convergence of the series. \( \sum_{n = 1}^{\infty} \frac {\sqrt{n}}{8^n} (x + 6)^n \)

Test the series for convergence or divergence. \( \displaystyle \sum_{n = 2}^{\infty} \frac {( - 1)^{n-1}}{{\sqrt {n} -1}} \)

Use the Ratio Test to determine whether the series is convergent or divergent. \( \displaystyle \sum_{n = 1}^{\infty} \frac {n!}{n^n} \)

Determine whether the series converges or diverges. \( \displaystyle \sum_{n = 1}^{\infty} \frac {2}{\sqrt n + 2} \)

Determine whether the series is convergent or divergent. \( \displaystyle \sum_{n = 1}^{\infty} \frac {1}{n^2 + 2n + 2} \)

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. \( 4 + 3 + \frac {9}{4} + \frac {27}{16} + \cdot \cdot \cdot \)

Find a formula for the general term \( a_n \) of the sequence, assuming that the pattern of the first few terms continues. \( \left\\{\begin{array} 1, 0, -1, 0, 1, 0, -1, 0, . . . .\end{array}\right\\} \)

(a) Approximate \( f \) by a Taylor polynomial with degree \( n \) at the number \( a. \) (b) Use Taylor's Inequality to estimate the accuracy of the approximation \( f(x) \approx T_n(x) \) when \( x \) lies in the given interval. (c) Check you result in part (b) by graphing \( \mid R_n(x) \mid . \) \( f (x) = e^{x^{2}}, \) \( a = 0, \) \( n = 3, \) \( 0 \le x \le 0.1 \)

Find the Taylor series for \( f(x) \) centered at the given value of \( a. \) [Assume that \( f \) has a power series expansion. Do not show that \( R_n (x) \to 0.\)] Also find the associated radius of convergence. \( f(x) = x^5 + 2x^3 + x, \) \(a = 2 \)

Find a power series representation for the function and determine the radius of convergence. \( f(x) = \frac {1 + x}{(1 - x)^2} \)

Use the Ratio Test to determine whether the series is convergent or divergent. \( \displaystyle \sum_{n = 1}^{\infty} \frac {n^{100} 100^n}{n!} \)

Test the series for convergence or divergence. \( \displaystyle \sum_{n = 1}^{\infty} (-1)^n \frac {n^n}{n!} \)

Determine whether the series converges or diverges. \( \displaystyle \sum_{n = 1}^{\infty} \frac {n + 1}{n^3 + n} \)

Determine whether the series is convergent or divergent. \( \displaystyle \sum_{n = 1}^{\infty} \frac {n^3}{n^4 + 4} \)

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. \( 10 - 2 + 0.4 - 0.08 + \cdot \cdot \cdot \)

Calculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence by hand. Does the sequence appear to have a limit? If so, calculate it. If not, explain why. \( a_n = \frac {3n}{1 + 6n} \)

Find the radius of convergence and interval of convergence of the series. \( \sum_{n = 1}^{\infty} \frac {(x - 2)^n}{n^n} \)

(a) Approximate \( f \) by a Taylor polynomial with degree \( n \) at the number \( a. \) (b) Use Taylor's Inequality to estimate the accuracy of the approximation \( f(x) \approx T_n(x) \) when \( x \) lies in the given interval. (c) Check you result in part (b) by graphing \( \mid R_n(x) \mid . \) \( f (x) = x \ln x, \) \( a = 1, \) \( n = 3, \) \( 0.5 \le x \le 1.5 \)

Find the Taylor series for \( f(x) \) centered at the given value of \( a. \) [Assume that \( f \) has a power series expansion. Do not show that \( R_n (x) \to 0.\)] Also find the associated radius of convergence. \( f(x) = x^6 - x^4 + 2, \) \( a = -2 \)

Find a power series representation for the function and determine the radius of convergence. \( f(x) = \frac {x^2 + x}{(1 - x)^3} \)

Find the radius of convergence and interval of convergence of the series. \( \sum_{n = 1}^{\infty} \frac {(2x - 1)^n}{5^ \sqrt{n}} \)

Test the series for convergence or divergence. \( \displaystyle \sum_{k = 1}^{\infty} \frac {\sqrt [3]{k} - 1}{k (\sqrt{k} + 1)} \)

Use the Ratio Test to determine whether the series is convergent or divergent. \( \displaystyle \sum_{n = 1}^{\infty} \frac {(2n)!}{(n!)^2} \)

Determine whether the series converges or diverges. \( \displaystyle \sum_{n = 1}^{\infty} \frac {n^2 + n + 1}{n^4 + n^2} \)

Determine whether the series is convergent or divergent. \( \displaystyle \sum_{n = 3}^{\infty} \frac {3n - 4}{n^2 - 2n} \)

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. \( 2 + 0.5 + 0.125 + 0.03125 + \cdot \cdot \cdot \)

Calculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence by hand. Does the sequence appear to have a limit? If so, calculate it. If not, explain why. \( a_n = 2 + \frac {(-1)^n}{n} \)

(a) Approximate \( f \) by a Taylor polynomial with degree \( n \) at the number \( a. \) (b) Use Taylor's Inequality to estimate the accuracy of the approximation \( f(x) \approx T_n(x) \) when \( x \) lies in the given interval. (c) Check you result in part (b) by graphing \( \mid R_n(x) \mid . \) \( f (x) = x \sin x, \) \( a = 0, \) \( n = 4, \) \( - 1 \le x \le 1 \)