$(0)$ If the series is convergent, $($i$)$ find a partial sum to approximate the sum of the series so that the error is guaranteed to be less than $0\mathrm{.}01.($ii$)$ Find an upper estimate and a lower estimate for the sum of the series that both have an error of less than $0\mathrm{.}01.$

${\displaystyle \underset{n=1}{\overset{}{}}}(-1{)}^{n-1}\frac{n}{n^2+1}$