Math 126B

Worksheet #3

Spring 2003

Names:

1. In an attempt to get you prepared for some major event coming up in a week or two, this

worksheet provides you with practice of problems you might see again. Discuss with your

group what test or tests to apply to each series. Then work individually to determine whether

each series converges or diverges. Compare results with your group.

(a) ${\displaystyle \sum }_{\mathit{k}\mathrm{=}\mathrm{1}}^{\mathrm{\infty }}\mathrm{(}\mathrm{-}\mathrm{1}{\mathrm{)}}^{\mathit{k}}\frac{{\mathit{k}}^{\mathrm{3}}\mathrm{+}\mathit{k}}{{\mathit{k}}^{\mathrm{6}}\mathrm{-}{\mathit{k}}^{\mathrm{2}}\mathrm{+}\mathrm{9}}$

(b) ${\displaystyle \sum }_{\mathit{n}\mathrm{=}\mathrm{1}}^{\mathrm{\infty }}\mathrm{(}\mathrm{-}\mathrm{1}{\mathrm{)}}^{\mathit{n}}\arctan \mathrm{(}\mathit{n}\mathrm{)}$

(c) ${\displaystyle \sum }_{\mathit{j}\mathrm{=}\mathrm{1}}^{\mathrm{\infty }}\mathrm{(}\mathrm{-}\mathrm{1}{\mathrm{)}}^{\mathit{j}}\frac{\mathit{j}}{{\mathit{j}}^{\mathrm{2}}\mathrm{+}\mathrm{1}}$

(d) ${\displaystyle \sum }_{\mathit{k}\mathrm{=}\mathrm{1}}^{\mathrm{\infty }}\frac{\mathrm{(}\mathrm{-}\mathrm{1}{\mathrm{)}}^{\mathit{k}\sqrt{\mathit{k}}}}{\mathit{k}\mathrm{+}\mathrm{1}\mathrm{2}}$

(there's more on the back...)

Math 126B

Worksheet #3

Spring 2003

Problem 1, continued

(e) ${\displaystyle \sum }_{\mathit{m}\mathrm{=}\mathrm{1}}^{\mathrm{\infty }}\frac{{\mathit{m}}^{\mathrm{3}}}{\mathit{m}\mathrm{!}}$

(f) ${\displaystyle \sum }_{\mathit{n}\mathrm{=}\mathrm{1}}^{\mathrm{\infty }}\frac{\mathit{n}\mathrm{!}}{{\mathrm{2}}^{\mathit{n}}}$

2. If you have time, here are problems for additional practice. Determine whether each of the

following converges absolutely, converges conditionally, or diverges.

(a) ${\displaystyle \sum }_{\mathit{j}\mathrm{=}\mathrm{1}}^{\mathrm{\infty }}\frac{\mathrm{3}\mathrm{ċ}\mathrm{5}\mathrm{.}\mathrm{ċ}\mathrm{\text{...}}\mathrm{\text{...}}\mathrm{(}\mathrm{2}\mathit{j}\mathrm{+}\mathrm{1}\mathrm{)}}{\mathrm{2}\mathrm{4}\mathrm{ċ}\mathrm{ċ}\mathrm{\left(}\mathrm{2}\mathit{j}\mathrm{\right)}}$.

(b) ${\displaystyle \sum }_{\mathit{k}\mathrm{=}\mathrm{1}}^{\mathrm{\infty }}\frac{\mathrm{(}\mathrm{1}\mathrm{+}\mathrm{(}\mathrm{-}\mathrm{1}{\mathrm{)}}^{\mathit{k}}{\mathrm{)}}^{\mathit{k}}}{{\mathit{e}}^{\mathit{k}}}$

(c) ${\displaystyle \sum }_{\mathit{n}\mathrm{=}\mathrm{1}}^{\mathrm{\infty }}\frac{\mathrm{(}\mathrm{-}\mathrm{1}{\mathrm{)}}^{\mathit{n}}{\mathit{n}}^{\mathrm{3}}}{\mathrm{(}\mathit{n}\mathrm{+}\mathrm{1}\mathrm{)}\mathrm{!}}$