Mathematics

Mathematics

Problem

For the series ∑k=1∞(−1)k+1xkk,0<x≤1,\sum_{k=1}^{\infty}(-1)^{k+1}\frac{x^{k}}{k}, \qquad 0<x\le 1, write the inequality that must hold between consecutive term magnitudes to confirm that the terms form a decreasing sequence, and state that inequality explicitly.
Intuitive explanation

The Alternating Series Test hinges on whether the absolute values of terms shrink monotonically—that's what guarantees convergence. Here, you need to verify that each term's magnitude is smaller than the previous one, which means comparing xkk\frac{x^k}{k} to xk−1k−1\frac{x^{k-1}}{k-1}. The key insight is that this inequality will depend on the relationship between xx and the ratio of consecutive indices.

Intuitive explanation

The Alternating Series Test hinges on whether the absolute values of terms shrink monotonically—that's what guarantees convergence. Here, you need to verify that each term's magnitude is smaller than the previous one, which means comparing xkk\frac{x^k}{k} to xk−1k−1\frac{x^{k-1}}{k-1}. The key insight is that this inequality will depend on the relationship between xx and the ratio of consecutive indices.

Browse the problem library