Chapter 11

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 \( f, \) and graph \( f \) and several partial sums \( s_n(x) \) on the same screen. What happens as \( n \) increases? \( f(x) = \frac {x^2}{x^2 + 1} \)

Find the radius of convergence and interval of convergence of the series. \( \sum_{n = 1}^{\infty} \frac {n}{b^n} (x - a)^n, b > 0 \)

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

Use the Ratio Test to determine whether the series is convergent or divergent. \( 1 - \frac {2!}{1 \cdot 3} + \frac {3!}{1 \cdot 3 \cdot 5} - \frac {4!}{1 \cdot 3 \cdot 5 \cdot 7} + \cdot \cdot \cdot \) \( ( - 1 )^{n-1} \frac {n!}{1 \cdot 3 \cdot 5 \cdot \space \cdot \cdot \cdot \space \cdot (2n - 1)} + \cdot \cdot \cdot \)

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

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

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. \( \displaystyle \sum_{n = 1}^{\infty} 12(0.73)^{n-1} \)

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. \( a_n = 1 + (- \frac {1}{2})^n \)

Graph both the sequence of terms and the sequence of partial sums on the same screen. Use the graph to make a rough estimate of the sum of the series. Then use the Alternating Series Estimation Theorem to estimate the sum correct to four decimal places. \( \displaystyle \sum_{n = 1}^{\infty} \frac {(-0.8)^n}{n!} \)

(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 \( \mid R_n(x) \mid . \) \( f (x) = \sinh 2x, \) \( a = 0, \) \( n = 5, \) \( - 1 \le x \le 1 \)

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. \( f(x) = 1/x,\) \( a = -3 \)

Find a power series representation for \( f, \) and graph \( f \) and several partial sums \( s_n(x) \) on the same screen. What happens as \( n \) increases? \( f(x) = \ln (1 + x^4) \)

Find the radius of convergence and interval of convergence of the series. \( \sum_{n = 1}^{\infty} \frac {b^n}{\ln n} (x - a)^n, b > 0 \)

Test the series for convergence or divergence. \( \displaystyle \sum_{k = 1}^{\infty} \frac {1}{2 + \sin k} \)

Determine whether the series converges or diverges. \( \displaystyle \sum_{n = 3}^{\infty} \frac {n + 2}{(n + 1)^3} \)

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

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. \( \displaystyle \sum_{n = 1}^{\infty} \frac {5}{\pi^n} \)

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. \( a_n = 1 + \frac{10^n}{9^n} \)

Use the Ratio Test to determine whether the series is convergent or divergent. \( \frac {2}{3} + \frac {2 \cdot 5}{3 \cdot 5} + \frac {2 \cdot 5 \cdot 8}{3 \cdot 5 \cdot 7} + \frac {2 \cdot 5 \cdot 8 \cdot 11}{3 \cdot 5 \cdot 7 \cdot 9} + \cdot \cdot \cdot \)

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. \( f(x) = e^{2x},\) \( a = 3 \)

Find a power series representation for \( f, \) and graph \( f \) and several partial sums \( s_n(x) \) on the same screen. What happens as \( n \) increases? \( f(x) = \ln \left( \frac {1 + x}{1 - x} \right) \)

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

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

Use the Ratio Test to determine whether the series is convergent or divergent. \( \displaystyle \sum_{n = 1}^{\infty} \frac {2 \cdot 4 \cdot 6 \cdot \space \cdot \cdot \cdot \space \cdot (2n)}{n!} \)

Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to the indicated accuracy? \( \displaystyle \sum_{n = 1}^{\infty} \frac{( - 1)^{n+1}}{n^6} (|error| < 0.00005) \)

Determine whether the series converges or diverges. \( \displaystyle \sum_{n = 1}^{\infty} \frac {5 + 2n}{(1 + n^2)^2} \)

Determine whether the series is convergent or divergent. \( \displaystyle \sum_{k = 1}^{\infty} ke^{-k} \)

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. \( \displaystyle \sum_{n = 1}^{\infty} \frac {(-3)^{n -1}}{4^n} \)

Determine whether the sequence converges or diverges. If it converges, find the limit. \( a_n = \frac {3 + 5n^2}{n + n^2} \)

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. \( f(x) = \cos x, \) \( a = \pi/2 \)

Find a power series representation for \( f, \) and graph \( f \) and several partial sums \( s_n(x) \) on the same screen. What happens as \( n \) increases? \( f(x) = \tan^{-1} (2x) \)

Find the radius of convergence and interval of convergence of the series. \( \sum_{n = 1}^{\infty} \frac {n^2x^n}{2 \cdot 4 \cdot 6 \cdot \cdot \cdot \cdot \cdot (2n)} \)

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

Use the Ratio Test to determine whether the series is convergent or divergent. \( \displaystyle \sum_{n = 1}^{\infty} ( - 1 )^n \frac {2^n n!}{5 \cdot 8 \cdot 11 \cdot \space \cdot \cdot \cdot \space (3n + 2 )} \)

Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to the indicated accuracy? \( \displaystyle \sum_{n = 1}^{\infty} \frac {( - \frac {1}{3})^n} {n} (|error| < 0.0005) \)

Determine whether the series converges or diverges. \( \displaystyle\sum_{n = 1}^{\infty} \frac {n + 3^n}{n+ 2^n} \)

Determine whether the series is convergent or divergent. \( \displaystyle \sum_{k = 1}^{\infty} ke^{-k^2} \)

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. \( \displaystyle \sum_{n = 0}^{\infty} \frac {3^{n + 1}}{(-2)^n} \)

Determine whether the sequence converges or diverges. If it converges, find the limit. \( a_n = \frac {3 + 5n^2}{1 + n} \)

Use Taylor's Inequality to determine the number of terms of the Maclaurin series for \( e^x \) that should be used to estimate \( e^{0.1} \) to within \(0.00001.\)

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. \( f(x) = \sin x, \) \( a = \pi \)

Evaluate the indefinite integral as a power series. What is the radius of convergence? \( \int \frac {t}{1 - t^8} dt \)

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

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

Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to the indicated accuracy? \( \displaystyle \sum_{n = 1}^{\infty} \frac {( - 1)^{n-1}}{n^2 2^n} (|error| < 0.0005) \)

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

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

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. \( \displaystyle \sum_{n = 1}^{\infty} \frac {e^{2n}}{6^{n - 1}} \)

Determine whether the sequence converges or diverges. If it converges, find the limit. \( a_n = \frac {n^4}{n^3 - 2n} \)