The series \(\displaystyle \sum_{n=1}^\infty \frac{(-1)^{n+1}}{\sqrt{n}}\) converges to \(S\). Based on the alternating series error bound, what is the least number of terms in the series that must be summed to gurantee a partial sum that is within 0.03 of \(S\) ?

Summary Submit

The absolute value of the maximum error can be obtained by obtaining the \(a_{n+1}\) term. In other words, if we used \(n\) terms to obtain a sum of \(S\), we can find that the error is within \(0.03\) of \(S\) by obtaining the \(n+1\) term.

$$ a_{n+1} = \frac{1}{\sqrt{n+1}} \lt 0.03 $$ $$ \sqrt{n+1} \gt \frac{1}{0.03} $$ $$ n \gt 1110.11 $$