Chapter 8: Series

Define the term power series.

Determine whether the sequence is convergent or divergent. If it is convergent, find its limit \({{\rm{a}}_{\rm{n}}}{\rm{ = }}\frac{{{\rm{2 + }}{{\rm{n}}^{\rm{3}}}}}{{{\rm{1 + 2}}{{\rm{n}}^{\rm{3}}}}}\).

Prove the series expansion of\(\sum\limits_{n = 0}^\infty {\frac{{{{( - 1)}^{n + 1}}}}{{(2n)!}}} {\left( {x - \frac{\pi }{2}} \right)^{2n}}\)represents\(\sin x\) for all\(x{\rm{. }}\)

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

\(\sum\limits_{n = 1}^\infty {\frac{{{{( - 2)}^n}}}{{{n^2}}}} \)

:\(\sum\limits_{n = 1}^\infty {\left( {{{(0.8)}^{n - 1}} - {{(0.3)}^n}} \right)} \) Find Whether It Is Convergent Or Divergent. If It Is Convergent Find Its Sum.

Find the radius of convergence and interval of convergence of the series \(\sum\limits_{n = 1}^\infty {\frac{{{{(2x - 1)}^n}}}{{{5^n}\sqrt n }}} \)

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

\({a_n} = \cos \left( {\frac{2}{n}} \right)\)

To find the power series representation for the function for the function \(f(x) = \frac{{{x^2} + x}}{{{{(1 - x)}^3}}}\) and determine the radius of convergence.

Determine whether the series is convergent or divergent.

\(\sum\limits_{{\rm{n = 1}}}^\infty {\frac{{\sqrt {{\rm{n + 1}}} {\rm{ - }}\sqrt {{\rm{n - 1}}} }}{{\rm{n}}}} \)

To determine the power series representation for the function \(f(x) = \frac{x}{{{x^2} + 16}}\) and graph \(f(x)\) and several partial sums \({s_n}(x)\).