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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?

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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 $$\frac{1}{(N+1)^5}\le \frac{1}{1000000}$$

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If you do not see how to solve the inequality, I will continue.

\begin{equation} \frac{1}{(N+1)^5}\le \frac{1}{10^6}\\ \end{equation} \begin{equation} (N+1)^5\ge10^6 \end{equation} \begin{equation} N+1\ge10\sqrt{10} \end{equation} \begin{equation} N\ge31 \end{equation}

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