If [math]\sum a_k[/math] convergences, then [math]lim_{k\longrightarrow\infty}a_k=0.[/math]  If the limit does not equal 0, then the series diverges.

The Harmonic Series [math]\sum_{k=1}^{\infty}\frac{1}{k}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+.....[/math] diverges even though the terms approach zero

 Suppose f is a continuous, positive, and decreasing function for [math]x\ge1[/math], and let [math]a_k=f\left(k\right)[/math] for k= 1, 2, 3, 4.... Then[br][math]\sum_{k=1}^{\infty}a_k[/math] and [math]\int_1^{\infty}f\left(x\right)dx[/math][br]either both converge or both diverge. In the case of convergence, the value of the integral is not equal to the value of the series

The p-series [math]\sum_{k=1}^{\infty}\frac{1}{k^p}[/math] converges for [math]p>1[/math] and diverges for [math]p\le1[/math]

Suppose [math]\sum a_k[/math] converges to A and [math]\sum b_k[/math] converges to b. Then[br]A) [math]\sum ca_k=c\sum a_k=cA[/math][br]B) [math]\sum\left(a_k\pm b_k\right)=\sum a_k\pm\sum b_k=A\pm B[/math]