\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
For the alternating harmonic series \(\sum_{n \ge 1} (-1)^n / n\), give the sign of the remainder \(R_N\) (for \(N \ge 1\)) and prove the bound \(|R_N| \le 1/(N+1)\).
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