65
Question
Estimate the value of the convergent series
\[
\sum_{n=1}^{\infty} \frac{6(-1)^{n}}{n^{4}+5}
\]
with an absolute error less than 0.001 .
Round your answer to six decimal places if necessary.

Step 1 ：The given series is an alternating series. We can use the Alternating Series Estimation Theorem to estimate the value of the series. The theorem states that the absolute error is less than or equal to the first neglected term. So we need to find the smallest n such that the absolute value of the nth term is less than 0.001.

Step 2 ：We have found that the 9th term of the series is the first term that is less than 0.001. Therefore, we can estimate the value of the series by summing the first 8 terms.

Step 3 ：The sum of the first 8 terms of the series is approximately -0.7670073306696904.

Step 4 ：Rounding to six decimal places, the estimated value of the convergent series is \(\boxed{-0.767007}\).