Final Answer:
The estimated value of the convergent series ∑ₖ=1^[∞](-1)ₖ/(k+2) with an absolute error less than 10⁻³ is approximately -0.694.
Step-by-step explanation:
To estimate the value of the given series, we can use the Alternating Series Estimation Theorem, which states that the absolute error of an alternating series is less than or equal to the absolute value of the first omitted term. For the series ∑ₖ=1^[∞](-1)ₖ/(k+2), we consider the absolute value of the term when k = 3, which is 1/5.
Now, we need to ensure that the absolute error is less than 10⁻³. Therefore, the series can be estimated as the sum of the first three terms (-1/3 + 1/4 - 1/5). Calculating this gives us an estimated value of approximately -0.694.
This estimation guarantees that the absolute error is within the specified limit. By truncating the series after three terms, we obtain a sufficiently accurate approximation while maintaining computational efficiency. It is crucial to recognize that this approach is applicable to convergent alternating series, and the theorem provides a reliable means of estimating their values with controlled error bounds.