Chapter 11

Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent? \( \displaystyle \sum_{n = 1}^{\infty} \frac {1}{n^4 + n^2} \)

List the first five terms of the sequence. \( a_n = \frac {(-1)^{n-1}}{5^n} \)

Find the Taylor polynomials \( T_3(x) \) for the function \( f \) centered at the number \( a \) Graph \( f \) and \( T_3 \) on the same screen. \( f(x) = e^{-x} \sin x, \) \( a = 0 \)

Use the definition of a Taylor series to find the first four nonzero terms of the series for \( f(x) \) centered at the given value of \( a. \) \( f(x) = \frac {1}{1 + x}, \) \( a = 2 \) $

Find a power series representation for the function and determine the interval of convergence. \( f(x) = \frac {4}{2x + 3} \)

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

Test the series for convergence or divergence. \( \displaystyle \sum_{n = 1}^{\infty} \frac {n^{2n}}{(1 + n)^{3n}} \)

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

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

Use the Integral Test to determine whether the series is convergent or divergent. \( \displaystyle \sum_{n = 1}^{\infty} \frac {1}{\left(3n - 1 \right)^34} \)

Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent? \( \displaystyle \sum_{n = 1}^{\infty} \frac {1}{\sqrt [3]{n}} \)

List the first five terms of the sequence. $$ a_n = \cos {n \pi}{2} $$

Find the Taylor polynomials \( T_3(x) \) for the function \( f \) centered at the number \( a \) Graph \( f \) and \( T_3 \) on the same screen. \( f(x) = \ln x, \) \( a = 1 \)

Use the definition of a Taylor series to find the first four nonzero terms of the series for \( f(x) \) centered at the given value of \( a. \) $$ f(x) = \sqrt [3]{x}, \quad a = 8 $$

Find a power series representation for the function and determine the interval of convergence. \( f(x) = \frac {x^2}{x^4 + 16} \)

Find the radius of convergence and interval of convergence of the series. \( \sum_{n = 0}^{\infty} \frac {x^n}{n!} \)

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

Use the Ratio Test to determine whether the series is convergent or divergent. \( \displaystyle \sum_{n = 1}^{\infty} \frac {n}{5^n} \)

Determine whether the series converges or diverges. \( \displaystyle \sum_{n = 1}^{\infty} \frac {9^n}{3 + 10^n} \)

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

List the first five terms of the sequence. \( a_n = \frac {1}{n + 1}! \)

Find the Taylor polynomials \( T_3(x) \) for the function \( f \) centered at the number \( a \) Graph \( f \) and \( T_3 \) on the same screen. \( f(x) = x \cos x, \) \( a = 0 \)

Use the definition of a Taylor series to find the first four nonzero terms of the series for \( f(x) \) centered at the given value of \( a. \) \( f(x) = \ln x, \) \( a = 1 \)

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

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

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

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

Determine whether the series converges or diverges. \( \displaystyle \sum_{n = 1}^{\infty} \frac {6^n}{5^n - 1} \)

Use the Integral Test to determine whether the series is convergent or divergent. \( \displaystyle \sum_{n = 1}^{\infty} n^2 e^{-n^3} \)

Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent? \( \displaystyle \sum_{n = 1}^{\infty} \frac {(-1)^{n - 1}}{n!} \)

List the first five terms of the sequence. \( a_n = \frac {(-1)^nn}{n! + 1} \)

Find the Taylor polynomials \( T_3(x) \) for the function \( f \) centered at the number \( a \) Graph \( f \) and \( T_3 \) on the same screen. \( f(x) = xe^{-2x}, \) \( a = 0 \)

Use the definition of a Taylor series to find the first four nonzero terms of the series for \( f(x) \) centered at the given value of \( a. \) \( f(x) = \sin x, \) \( a = \pi/6 \)

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

Find the radius of convergence and interval of convergence of the series. \( \sum_{n = 1}^{\infty} \frac {x^n}{n^44^n} \)

Test the series for convergence or divergence. \( \displaystyle \sum_{n = 0}^{\infty} ( - 1)^n \frac {\pi^{2n}}{(2n)!} \)

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

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

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

Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why. \( \displaystyle \sum_{n = 1}^{\infty} \frac {12}{(-5)^n} \)

List the first five terms of the sequence. \( a_1 = 1, a_{n+1} = 5a_n - 3 \)

Find the Taylor polynomials \( T_3(x) \) for the function \( f \) centered at the number \( a \) Graph \( f \) and \( T_3 \) on the same screen. \( f(x) = \tan^{-1} x, \) \( a = 1 \)

Use the definition of a Taylor series to find the first four nonzero terms of the series for \( f(x) \) centered at the given value of \( a. \) $$ f(x) = \cos^2 x, \quad a = 0 $$

Find a power series representation for the function and determine the interval of convergence. \( f(x) = {x + a}{x^2 + a^2}, a > 0 \)

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

Use the Ratio Test to determine whether the series is convergent or divergent. \( \displaystyle \sum_{n = 0}^{\infty} \frac {( - 3)^n}{(2n + 1)!} \)

Test the series for convergence or divergence. \( \displaystyle \sum_{n = 1}^{\infty} (-1)^n \frac {\sqrt{n}}{2n + 3} \)

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

Determine whether the series is convergent or divergent. \( \displaystyle \sum_{n = 3}^{\infty} n^{-0.9999} \)

List the first five terms of the sequence. \( a_1 = 6, a_{n+1} = \frac {a_n}{n} \)