# How would you solve this alternating series problem using the remainder formula?

Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than $10^{-6}$. $$\sum_{k=1}^{\infty}\frac{{}(-1)^k}{k^5}$$

I am unsure of what steps to take to solve this problem. How is the alternate series test related to the problem?

In an alternating series the error in stopping the sum at the $N$th term is less than or equal to the absolute value of the $(N+1)$st term, so you want to solve $$\frac{1}{(N+1)^5}\le \frac{1}{1000000}$$
$$\frac{1}{(N+1)^5}\le \frac{1}{10^6}\\$$ $$(N+1)^5\ge10^6$$ $$N+1\ge10\sqrt{10}$$ $$N\ge31$$