Chapter 11

Find the radius of convergence and interval of convergence of the series. \( \sum_{n = 1}^{\infty} \frac {5x - 4)^n}{n^3} \)

How many terms of the Maclaurin series for \( \ln(1 + x) \) do you need to use to estimate \( \ln \) 1.4 to within 0.001?

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. \( f(x) = \sqrt x \) \( a = 16 \)

Evaluate the indefinite integral as a power series. What is the radius of convergence? \( \int \frac {t}{1 + t^3} dt \)

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

Test the series for convergence or divergence. \( \displaystyle \sum_{n = 1}^{\infty} \frac {n^2 + 1}{5^n} \)

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

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? \( \displaystyle \sum_{n = 1}^{\infty} \left( - \frac {1}{n} \right )^n (|error| < 0.00005) \)

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

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

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. \( \displaystyle \sum_{n = 1}^{\infty} \frac {6 \cdot 2^{2n - 1}}{3^n} \)

Determine whether the sequence converges or diverges. If it converges, find the limit. \( a_n = 2 + (0.86)^n \)

Evaluate the indefinite integral as a power series. What is the radius of convergence? \( \int x^2 \ln (1 + x) dx \)

Find the radius of convergence and interval of convergence of the series. \( \sum_{n = 1}^{\infty} \frac {x^n}{1 \cdot 3 \cdot 5 \cdot \cdot \cdot\cdot \cdot (2n - 1)} \)

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

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

Approximate the sum of the series correct to four decimal places. \( \displaystyle \sum_{n = 1}^{\infty} \frac {( - 1)^n}{(2n)!} \)

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

Explain why the Integral Test can't be used to determine whether the series is convergent. \( \displaystyle \sum_{n = 1}^{\infty} \frac {\cos \pi n}{\sqrt n} \)

Determine whether the sequence converges or diverges. If it converges, find the limit. \( a_n = 3^n 7^{-n} \)

Use the Alternating Series Estimation Theorem or Taylor's Inequality to estimate the range of values of \( x \) for which the given approximation is accurate to within the stated error. Check your answer graphically. \( \sin x \approx x - \frac {x^3}{6} \) \( (\mid error \mid < 0.001) \)

Determine whether the series is convergent or divergent. If it is convergent, find its sum. \( \frac {1}{3} + \frac {1}{6} + \frac {1}{9} + \frac {1}{12} + \frac {1}{15} + \cdot \cdot \cdot \)

Use the Alternating Series Estimation Theorem or Taylor's Inequality to estimate the range of values of \( x \) for which the given approximation is accurate to within the stated error. Check your answer graphically. \( \cos x \approx 1 - \frac {x^2}{2} + \frac {x^4}{24} \) \( (\mid error \mid < 0.005) \)

Find the radius of convergence and interval of convergence of the series. \( \sum_{n = 1}^{\infty} \frac {n!x^n}{1 \cdot 3 \cdot 5 \cdot \cdot \cdot \cdot \cdot (2n - 1)} \)

Use the Root Test to determine whether the series is convergent or divergent. \( \displaystyle \sum_{n = 1}^{\infty} \left( \frac { - 2n}{n + 1} \right)^{5n} \)

Approximate the sum of the series correct to four decimal places. \( \displaystyle \sum_{n = 1}^{\infty} \frac {( - 1 )^{n+1}}{n^6} \)

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

Explain why the Integral Test can't be used to determine whether the series is convergent. \( \displaystyle \sum_{n = 1}^{\infty} \frac {\cos^2 n}{1 + n^2} \)

Determine whether the series is convergent or divergent. If it is convergent, find its sum. \( \frac {1}{3} + \frac {2}{9} + \frac {1}{27} + \frac {2}{81} + \frac {1}{243} + \frac {2}{729} + \cdot \cdot \cdot \)

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

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

Use the Alternating Series Estimation Theorem or Taylor's Inequality to estimate the range of values of \( x \) for which the given approximation is accurate to within the stated error. Check your answer graphically. \( \arctan x \approx x - \frac {x^3}{3} + \frac {x^5}{5} \) \( (\mid error \mid < 0.05) \)

Use a power series to approximate the definite integral to six decimal places. \( \int^{0.3}_0 \frac {x}{1 + x^3} dx \)

If \( \sum_{n = 0}^{\infty} c_n4^n \) is convergent, can we conclude that each of the following series is convergent? (a) \( \sum_{n = 0}^{\infty} c_n ( - 2)^n \) (b) \( \sum_{n = 0}^{\infty} c_n ( - 4)^n \)

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

Use the Root Test to determine whether the series is convergent or divergent. \( \displaystyle \sum_{n = 1}^{\infty} \left ( 1 + \frac {1}{n} \right)^{n2} \)

Approximate the sum of the series correct to four decimal places. \( \displaystyle \sum_{n = 1}^{\infty} ( - 1)^n ne^{-2n} \)

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

Find the values of \( p \) for which the series is convergent. \( \displaystyle \sum_{n = 2}^{\infty} \frac {{1}}{{n(\ln n)^P}} \)

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

Determine whether the sequence converges or diverges. If it converges, find the limit. \( a_n = e^{-1/ \sqrt n} \)

Suppose you know that \( f^{(n)}(4) = \frac {( - 1)^n n!}{3^n(n + 1)} \) and the Taylor series of \( f \) centered at 4 converges to \( f(x) \) for all \( x \) in the interval of convergence. Show that the fifthdegree Taylor polynomial approximates $ f(5) with error less than 0.0002.

Use a power series to approximate the definite integral to six decimal places. \( \int^{1/2}_0 \arctan (x/2) dx \)

Suppose that \( \sum_{n = 0}^{\infty} c_nx^n \) converges when \( x = - 4 \) and diverges when \( x = 6. \) What can be said about the convergence or divergence of the following series? (a) \( \sum_{n = 0}^{\infty} c_n \) (b) \( \sum_{n = 0}^{\infty} c_n8^n \) (c) \( \sum_{n = 0}^{\infty} c_n( - 3)^n \) (d) \( \sum_{n = 0}^{\infty} ( - 1)^n c_n 9^n \)

Test the series for convergence or divergence. \( \displaystyle \sum_{j = 1}^{\infty} ( - 1)^j \frac {\sqrt{j}}{j + 5} \)

Use the Root Test to determine whether the series is convergent or divergent. \( \displaystyle \sum_{n = 0}^{\infty} \) (arctan \( n)^n \)

Approximate the sum of the series correct to four decimal places. \( \displaystyle \sum_{n = 1}^{\infty} \frac {( - 1)^{n-1}}{n 4^n} \)

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

Determine whether the series is convergent or divergent. If it is convergent, find its sum. \( \displaystyle \sum_{k = 1}^{\infty} \frac {k^2}{k^2 - 2k + 5} \)

Determine whether the sequence converges or diverges. If it converges, find the limit. \( a_n = \sqrt { \frac {1 + 4n^2}{1 + n^2}} \)