Science:Infinite Series Module/Units/Unit 1/1.4 Properties of Convergent Series/1.4.01 The Properties of Convergent Series

The following properties may not come as a surprise to students, but are useful when determining whether more complicated series are convergent or divergent. Proofs of the theorem below can be found in most introductory Calculus textbooks and are relatively straightforward.

Theorem (Properties of Convergent Series)

If the two infinite series $\sum _{k=1}^{\infty }a_{k}\ ,\ \sum _{k=1}^{\infty }b_{k}$ are both convergent, and c is a real constant, then ${\begin{aligned}1)&\sum _{k=1}^{\infty }(a_{k}\pm b_{k})\ \mathrm {is\ convergent} \\2)&\sum _{k=1}^{\infty }ca_{k}\ \mathrm {is\ convergent} \\3)&\sum _{k=1}^{\infty }ca_{k}=c\sum _{k=1}^{\infty }a_{k}\\4)&\sum _{k=1}^{\infty }(a_{k}\pm b_{k})=\sum _{k=1}^{\infty }a_{k}\pm \sum _{k=1}^{\infty }b_{k}\end{aligned}}$