1. The problem asks for the least number of terms needed to approximate the alternating series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{\sqrt{n}}$ within an error of $\pm 0.001$ using the alternating series error test. 2. The alternating series error test states that the absolute error when approximating an alternating series by its first $N$ terms is less than or equal to the absolute value of the first omitted term: $$\left| R_N \right| \leq \left| a_{N+1} \right|$$ where $a_n = \frac{1}{\sqrt{n}}$ in this series. 3. To ensure the error is within $0.001$, we require: $$\left| a_{N+1} \right| = \frac{1}{\sqrt{N+1}} \leq 0.001$$ 4. Solve for $N+1$: $$\frac{1}{\sqrt{N+1}} \leq 0.001 \implies \sqrt{N+1} \geq \frac{1}{0.001} = 1000$$ 5. Square both sides: $$N+1 \geq 1000^2 = 1000000$$ 6. Subtract 1: $$N \geq 999999$$ 7. Therefore, the least number of terms needed is $N = 999999$. This means you must sum at least 999999 terms to approximate the series within $\pm 0.001$ error.