${\displaystyle \underset{n=1}{\overset{}{}}}\frac{(-1{)}^{n+1}(x-1{)}^n}{n}$

Use a graphing utility to graph several partial sums of the series.

$($a$)$ Find the sum of the series and its radius of convergence.

$($b$)$ Use a graphing utility and $50$ terms of the series to approximate the sum when $x=0\mathrm{.}2.($Round your answer to six decimal places.$)$

${\displaystyle \underset{n=1}{\overset{50}{}}}\frac{(-1{)}^{n+1}(-0\mathrm{.}8{)}^n}{n}$~~$1\mathrm{.}609438$

Determine what the approximation represents.

The sum from part $($b$)$ is an approximation of $1\mathrm{.}832581$

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Determine how good the approximation is$.($Round your answer to six decimal places.$)$ error $$