Convergence and Divergence Question 11

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 $S$ ?

34

333

1111

9999

Approach

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