Answer
We need the first five terms.
Thus, $\Sigma_{n=1}^{5}\frac{(-1)^{(n+1)}}{n^{6}}\approx 0.9856$
Work Step by Step
Alternating series test:
Suppose that we have a series $\Sigma a_n$, such that $a_{n}=(-1)^{n}b_n$ or $a_{n}=(-1)^{n+1}b_n$, where $b_n\geq 0$ for all $n$.
Then if the following two conditions are satisfied, the series is convergent.
1. $\lim\limits_{n \to \infty}b_{n}=0$
2. $b_{n}$ is a decreasing sequence.
In the given problem, $b_{n}=\frac{1}{n^{6}}$
which satisfies both conditions of the Alternating Series Test as follows:
1. $b_{n}=\frac{1}{n^{6}}$, is decreasing because the denominator is increasing.
2. $\lim\limits_{n \to \infty}b_{n}=\lim\limits_{n \to \infty}\frac{1}{n^{6}}=0$
Hence, the given series is convergent by the Alternating Series Test.
Now compute the terms $a_{n}=\frac{(-1)^{n+1}}{n^{6}}$ until we get to one where $|a_{n}|\lt 0.00005$
$a_{1}=1$
$a_{2}=-\frac{1}{64}=-0.015625$
$a_{3}=\frac{1}{729}\approx 0.00137174211$
$a_{4}=-\frac{1}{4096}\approx -0.00024414062$
$a_{5}=\frac{1}{15625}\approx 0.000064$
$a_{6}=-\frac{1}{46656}=-0.000021$
$|a_{6}|\lt 0.00005$
We need the first five terms to get the right accuracy.
Thus, $\Sigma_{n=1}^{5}\frac{(-1)^{(n+1)}}{n^{6}}\approx 0.9856$
