Suppose uk>0u_k>0uk>0. An alternating series is: ∑k=1n(−1)k−1uk=u1−u2+u3−u4+⋯\sum_{k=1}^n (-1)^{k - 1} u_k = u_1 - u_2 + u_3 - u_4 + \cdots k=1∑n(−1)k−1uk=u1−u2+u3−u4+⋯ Convergence test If ∀k uk>0\forall k\; u_k>0∀kuk>0, decreasing and limn→∞un=0\lim_{n\to \infty} u_n = 0limn→∞un=0, then: ∑k=1n(−1)k−1uk is converging\sum_{k=1}^n (-1)^{k - 1} u_k \text{ is converging} k=1∑n(−1)k−1uk is converging