Math 126B

Worksheet #4

Spring 2003

Names:

1. When working with power series, we Iind the interval of convergence. In order to Iind these

intervals, we have to set up an inequality using a particular value. Since $\mathrm{w}\mathrm{e}\mathrm{\text{'}}\mathrm{v}\mathrm{e}$ been working

with so many tests for convergence, many of the tests and their comparable values may be

getting all jumbled. This activity attempts to address this situation. In the following set of

problems, you will determine the limit of various expressions. You will explore how the same

result of the limits can lead to different interpretations.

(a) Determine whether the series ${\displaystyle \sum }_{\mathit{n}\mathrm{=}\mathrm{1}}^{\mathrm{\infty }}\frac{{\mathit{n}}^{\mathrm{2}}\mathrm{+}\mathrm{2}\mathit{n}\mathrm{-}\mathrm{1}}{\mathrm{3}{\mathit{n}}^{\mathrm{2}}\mathrm{-}\mathrm{6}}$ converges or diverges by Iinding the limit

of the sequence $\mathrm{\{}{\mathit{a}}_{\mathit{n}}\mathrm{=}\frac{{\mathit{n}}^{\mathrm{2}}\mathrm{+}\mathrm{2}\mathit{n}\mathrm{-}\mathrm{1}}{\mathrm{3}{\mathit{n}}^{\mathrm{2}}\mathrm{-}\mathrm{6}}\mathrm{\}}$.

(b) Use the Ratio Test to determine whether the series ${\displaystyle \sum }_{\mathit{n}\mathrm{=}\mathrm{1}}^{\mathrm{\infty }}\frac{{\mathit{n}}^{\mathrm{2}}\mathrm{-}\mathrm{2}}{{\mathrm{3}}^{\mathit{n}}}$ converges or diverges.

(c) Determine whether the series ${\displaystyle \sum }_{\mathit{n}\mathrm{=}\mathrm{1}}^{\mathrm{\infty }}\frac{{\mathrm{2}}^{\mathrm{-}\mathit{n}}\mathrm{(}{\mathit{n}}^{\mathrm{2}}\mathrm{+}\mathrm{2}\mathit{n}\mathrm{-}\mathrm{1}\mathrm{)}}{\mathrm{3}{\mathit{n}}^{\mathrm{2}}\mathrm{-}\mathrm{6}}$ converges or diverges by a limit

comparison to ${\displaystyle \sum }_{\mathit{n}\mathrm{=}\mathrm{1}}^{\mathrm{\infty }}{\mathrm{2}}^{\mathrm{-}\mathit{n}}$

(d) List below the tests for convergence that use 0, 1, and other numbers for comparison.

Compare to 0 Compare to 1 Compare to other numbers

Math 126B

Worksheet #4

Spring 2003

2. Use the the fact that

$\ln \mathrm{(}\mathrm{1}\mathrm{-}\mathit{x}\mathrm{)}\mathrm{=}\mathrm{-}{\displaystyle \sum }_{\mathit{n}\mathrm{=}\mathrm{1}}^{\mathrm{\infty }}\frac{{\mathit{x}}^{\mathit{n}}}{\mathit{n}}$ for $\mathrm{\left|}\mathit{x}\mathrm{\right|}\mathrm{<}\mathrm{1}$

to do the following problems.

(a) Express $\ln \mathrm{(}\mathrm{1}\mathrm{+}{\mathit{x}}^{\mathrm{3}}\mathrm{/}\mathrm{2}\mathrm{)}$ as a power series centered at 0.

(b) Evaluate the indefinite integral ${\displaystyle \int }\ln \mathrm{(}\mathrm{1}\mathrm{+}{\mathit{x}}^{\mathrm{3}}\mathrm{/}\mathrm{2}\mathrm{)}\mathit{d}\mathit{x}$ as a power series.

(c) What is the radius and the interval of convergence of the series you found in part (b)?