The alternating series [math]\sum\left(-1\right)^{k+1}a_k[/math] converges provided [br]1) the terms of the series are monotonically decreasing [br]2) [math]lim_{k\longrightarrow\infty}a_k=0[/math]

Let [math]\sum_{k=1}^{\infty}\left(-1\right)^{k+1}a_k[/math] be a convergent alternating series with terms that are non increasing magnitude. Let [math]R_n=S-S_n[/math] be the remainder in approximating the value of the series by the sum of its first n terms. Then [math]R_n\le a_{n+1}[/math]. In other words, the remainder is less than or ewaul to the magnitude of the first neglected term.

If [math]\sum\left|a_k\right|[/math] converges, then [math]\sum a_k[/math] converges absolutely. If [math]\sum\left|a_k\right|[/math] diverges and [math]\sum a_k[/math] converges, then [math]\sum a_k[/math] converges conditionally.