QuestionQuestion 9 1 pts
Use the Alternating Series Estimation Theorem to find the minimum number of terms of the infinite series $\sum_{n=1}^{\infty} \frac{2(-1)^{n}}{n^{3}}$ we need to add to approximate the sum of the series with $\mid$ error $\mid \leq .016$. 8 10 4 6

Studdy Solution
Solve the inequality $\frac{2}{(n+1)^3} \leq 0.016$:
1. Multiply both sides by $(n+1)^3$: $ 2 \leq 0.016(n+1)^3 \]
2. Divide both sides by $0.016$: $ \frac{2}{0.016} \leq (n+1)^3 \]
3. Calculate $\frac{2}{0.016} = 125$.
4. Take the cube root of both sides: $ n+1 \geq \sqrt[3]{125} = 5 \]
5. Solve for $n$: $ n \geq 4 \]
Since $n$ must be an integer, the minimum $n$ is 4.
The minimum number of terms required is:
$$\boxed{4}$$