Answer
$0.9856$
Work Step by Step
Given: the series $\Sigma_{n=1}^\infty \dfrac{(-1)^{n+1}}{n^6}$
$S_1=\Sigma_{n=1}^\infty \dfrac{(-1)^{n+1}}{n^6}=\Sigma_{n=1}^3 \dfrac{(-1)^{n+1}}{n^6} \approx 9.85747$
$S_2=\Sigma_{n=1}^\infty \dfrac{(-1)^{n+1}}{n^6}=\Sigma_{n=1}^4 \dfrac{(-1)^{n+1}}{n^6} \approx 9.85502$
$S_3=\Sigma_{n=1}^\infty \dfrac{(-1)^{n+1}}{n^6}=\Sigma_{n=1}^5 \dfrac{(-1)^{n+1}}{n^6} \approx 9.85567$
$S_4=\Sigma_{n=1}^\infty \dfrac{(-1)^{n+1}}{n^6}=\Sigma_{n=1}^6 \dfrac{(-1)^{n+1}}{n^6} \approx 9.85545$
When approximated up to four decimals, our answer is: $0.9856$
