Chapter 8: Series

Determine whether the series is conditionally convergent, absolutely convergent or divergent.

Determine whether the series is absolutely convergent, conditionally convergent,or divergent.

\(\sum\limits_{n = 0}^\infty {\frac{{{{( - 3)}^n}}}{{(2n + 1)!}}} \)

The resistivity\(\rho \)of a conducting wire is the reciprocal of the conductivity and is measured in units of ohm-meters\((\Omega \$ - m)\). 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\(^\circ {\rm{C}}\). There are tables that list the values of\(\alpha \)(called the temperature coefficient) and\({\rho _{20}}\)(the resistivity at\({20^\circ }{\rm{C}}\)) 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 and\(\alpha = 0.0039{/^\circ }{\rm{C}}\) \({\rho _{20}} = 1.7 \times {10^{ - 8}}\Omega - {\rm{m}}\). Graph the resistivity of copper and the linear and quadratic approximations for\(0.014{\rm{gL}}\).

(c) For what values of\(t\)does the linear approximation agree with the exponential expression to within one percent.

To find the power series representation for the function \(f(x) = {\tan ^{ - 1}}(2x)\) and Sketch the graph \(f(x)\) and several partial sums \({s_n}(x)\).

Use the binomial series to expand the function as a power series. State the radius of convergence

Determine whether the sequence converges or diverges. If it converges, find the limit.

\({a_n} = \ln (n + 1) - \ln n\)

(a) Check whether the given series\(\sum\limits_{n = 0}^\infty {{c_n}} {x^n}\)is convergent or divergent.

(b) Check whether the given series\(\sum\limits_{n = 0}^\infty {{c_n}} {8^n}\)is convergent or divergent.

(c) Check whether the given series\(\sum\limits_{n = 0}^\infty {{c_n}} {( - 3)^n}\)is convergent or divergent.

(d) Check whether the given series\(\sum\limits_{n = 0}^\infty {{{( - 1)}^n}} {c_n}{9^n}\)is convergent or divergent.

Determine the radius of convergence of given series.

\(\sum\limits_{n = 0}^\infty {\frac{{{{(n!)}^k}}}{{(kn!)}}} {x^n}\)

Determine whether the series is absolutely convergent, conditionally convergent,or divergent.

\(\sum\limits_{k = 1}^\infty {k*{{(\frac{2}{3})}^k}} \)

Determine whether the sequence converges or diverges. If it converges, find the limit.

\({a_n} = \frac{{{{\cos }^2}n}}{{{2^n}}}\)