Chapter 11

Find the values of \( p \) for which the series is convergent. \( \displaystyle \sum_{n = 3}^{\infty} \frac {{1}}{{ n \ln n [\ln (\ln n)]^P}} \)

A car is moving with speed 20 m/s and acceleration 2 m/s^2 at a given instant. Using a second-degree Taylor polynomial, estimate how far the car moves in the next second. Would it be reasonable to use this polynomial to estimate the distance traveled during the next minute?

Use the binomial series to expand the function as a power series. State the radius of convergence. \( \sqrt [4]{1 - x} \)

Use a power series to approximate the definite integral to six decimal places. \( \int^{0.2}_0 x \ln (1 + x^2) dx \)

If \( k \) is a positive integer, find the radius of convergence of the series \( \sum_{n = 0}^{\infty} \frac {(n!)^k}{(kn)!} x^n \)

Test the series for convergence or divergence. \( \displaystyle \sum_{k= 1}^{\infty} \frac {5^k}{3^k + 4^k} \)

Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. \( \displaystyle \sum_{n = 2}^{\infty} \frac {( - 1)^n}{\ln n} \)

Is the 50th partial sum \( s_{50} \) of the alternating series \( \displaystyle \sum_{n = 1}^{\infty} (-1)^{n - 1} /n \) an overestimate or an underestimate of the total sum? Explain.

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

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

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

Determine whether the sequence converges or diverges. If it converges, find the limit. \( a_n = \frac {4^n}{1 + 9^n} \)

The resistivity \( \rho \) of a conducting wire is the reciprocal of the conductivity and is measured in units of ohm-meters \( (\Omega\text{-m}). \) The resistivity of a given metal depends on the temperature according to the equation \( \rho(t) = \rho_{20}e^{\alpha^{t-20}} \) where \( t \) is the temperature in \( ^oC. \) There are tables that list the values of (called the temperature coefficient) and \( \rho_{20} \) (the resistivity at \( 20^oC \) ) for various metals. Except at very low temperatures, the resistivity varies almost linearly with temperature and so it is common to approximate the expression for \( \rho(t) \) by its first- or second-degree Taylor polynomial at \( t = 20. \) (a) Find expressions for these linear and quadratic approximations. (b) For copper, the tables give \( \alpha = 0.0039/^oC \) and \( \rho_{20} = 1.7 x 10^{-8} \Omega\text{-m.} \). Graph the resistivity of copper and the linear and quadratic approximations for \( -250^oC \le t \le 1000^oC. \) (c) For what values of \( t \) does the linear approximation agree with the exponential expression to within one percent?

Use a power series to approximate the definite integral to six decimal places. \( \int^{0.3}_0 \frac {x^2}{1 + x^4} dx \)

Let \( p \) and \( q \) be real numbers with \( p < q. \) Find a power series whose interval of convergence is (a) \( (p, q) \) (b) \( (p, q] \) (c) \( [p, q) \) (d) \( [p, q] \)

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

Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. \( \displaystyle \sum_{n = 1}^{\infty} \left( \frac {1 - n}{2 + 3n} \right) ^n \)

For what value of \( p \) is each series convergent? \( \displaystyle \sum_{n = 1}^{\infty} \frac {( - 1)^{n-1}}{n^p} \)

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

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

Determine whether the series is convergent or divergent. If it is convergent, find its sum. $$ \displaystyle \sum_{n = 1}^{\infty} [(-0.2)^n + (0.06)^{n - 1}] $$

Determine whether the sequence converges or diverges. If it converges, find the limit. \( a_n = \cos \left( \frac {n \pi}{n + 1} \right) \)

Use the binomial series to expand the function as a power series. State the radius of convergence. \( \frac {1}{(2 + x)^3} \)

Is it possible to find a power series whose interval of convergence is \( [0, \infty)? \) Explain.?

Test the series for convergence or divergence. \( \displaystyle \sum_{n = 1}^{\infty} \left( \frac {n}{n + 1} \right)^{n2} \)

Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. \( \displaystyle \sum_{n = 1}^{\infty} \frac {( - 9)^n}{n10^{n+1}} \)

For what value of \( p \) is each series convergent? $$ \displaystyle \sum_{n = 1}^{\infty} \frac {( - 1)^n}{n + p} $$

Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error. $$ \displaystyle\sum_{n = 1}^{\infty} \frac {1}{5 + n^5} $$

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

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

Use the binomial series to expand the function as a power series. State the radius of convergence. \( (1 - x)^{3/4} \)

Show that the function \( f(x) = \sum_{n = 0}^{\infty} \frac {(-1)^n x^{2n}}{(2n)!} \) is a solution of the differential equation \( f''(x) + f(x) = 0 \)

Graph the first several partial sums \( s_n (x) \) of the series \( \sum_{n = 0}^{\infty} x^n, \) together with the sum function \( f(x) = 1/(1 - x), \) on a common screen. On what interval do these partial sums appear to be converging to \( f(x)? \)

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

Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. \( \displaystyle \sum_{n = 1}^{\infty} \frac {n5^{2n}}{10^{n+1}} \)

For what value of \( p \) is each series convergent? \( \displaystyle \sum_{n = 2}^{\infty} ( - 1)^{n-1} \frac {(\ln n)^P}{n} \)

Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error. \( \displaystyle \sum_{n = 1}^{\infty} \frac {e^{1/n}}{n^4} \)

Leonhard Euler was able to calculate the exact sum of the \( p- \) series with \( p = 2: \) \( \zeta (2) = \displaystyle \sum_{n = 1}^{\infty} \frac {1}{n^2} = \frac {\pi^2}{6} \) (See page 720.) Use this fact to find the sum of each series. (a) \( \displaystyle \sum_{n = 2}^{\infty} \frac {1}{n^2} \) (b) \( \displaystyle \sum_{n = 3}^{\infty} \frac {1}{(n + 1)^2} \) (c) \( \displaystyle \sum_{n = 1}^{\infty} \frac {1}{(2n)^2} \)

Determine whether the series is convergent or divergent. If it is convergent, find its sum. \( \displaystyle \sum_{n = 1}^{\infty} \frac {2^n + 4^n}{e^n} \)

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

The function \( J_1 \) defined by \( J_1(x) = \sum_{n = 0}^{\infty} \frac {(-1)^n x^{2n + 1}}{n! (n + 1)! 2^{2n + 1}} \) is called the Bessel function of order 1. (a) Find its domain. (b) Graph the first several partial sums on a common screen. (c) If your CAS has built-in Bessel Functions, graph \( J_1 \) on the same screen as the partial sums in part (b) and observe how the partial sums approximate \( J_1. \)

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

Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. \( \displaystyle \sum_{n = 2}^{\infty} \left( \frac {n}{\ln n} \right) ^n \)

Show that series \( \sum ( - 1)^{n-1}b_n, \) where \( b_n = 1/n \) if \( n \) is odd and \( b_n = 1/n^2 \) if \( n \) is even, is divergent. Why does the Alternating Series Test not apply?

Euler also found the sum of the \( p- \) series with \( p = 4: \) \( \zeta (4) = \displaystyle \sum_{n = 1}^{\infty} \frac {1}{n^4} = \frac {\pi^4}{90} \) Use Euler's result to find the sum of the series. (a) \( \displaystyle \sum_{n = 1}^{\infty} \left( \frac {3}{n} \right)^4 \) (b) \( \displaystyle \sum_{k = 5}^{\infty} \frac {1}{(k - 2)^4} \)

Determine whether the series is convergent or divergent. If it is convergent, find its sum. \( \displaystyle \sum_{k = 1}^{\infty} (\sin 100)^k \)

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

Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error. \( \displaystyle \sum_{n = 1}^{\infty} 5^{-n} \cos^2 n \)

The function \( A \) defined by $ A (x) = 1 + \frac {x^3}{2 \cdot 3} + \frac {x^6}{2 \cdot 3 \cdot 5 \cdot 6} + \frac {x^9}{\2 \cdot 3 \cdot 5 \cdot 6 \cdot 8 \cdot 9} + \cdot \cdot \cdot is called an Airy function after the English mathematician and astronomer Sir George Airy (1801- 1892). (a) Find the domain of the Airy function. (b) Graph the first several partial sums on a common screen. (c) If your CAS has built-in Airy functions, graph A on the same screen as the partial sums in part (b) and observe how the partial sums approximate A.

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