Science:Infinite Series Module/Units/Unit 1/1.1 Infinite Sequences/1.1.05 Limit Laws for Infinite Sequences

Convergent sequences have several properties that we can take advantage of. The proofs for the laws below are similar to those for the limit laws for functions, and as such are not provided.

Theorem: Limit Laws of Convergent Infinite Sequences

Suppose we are given two convergent infinite sequences $\{a_{n}\}_{n=1}^{\infty }$ and $\{b_{n}\}_{n=1}^{\infty }$ Then ${\begin{aligned}1)&\lim _{n\rightarrow \infty }(a_{n}\pm b_{n})=\lim _{n\rightarrow \infty }a_{n}\pm \lim _{n\rightarrow \infty }b_{n}\\2)&\lim _{n\rightarrow \infty }ca_{n}=c\lim _{n\rightarrow \infty }a_{n}\\3)&\lim _{n\rightarrow \infty }a_{n}b_{n}=\lim _{n\rightarrow \infty }a_{n}\cdot \lim _{n\rightarrow \infty }b_{n}\\4)&\lim _{n\rightarrow \infty }{\frac {a_{n}}{b_{n}}}={\frac {\lim _{n\rightarrow \infty }a_{n}}{\lim _{n\rightarrow \infty }b_{n}}},\quad \lim _{n\rightarrow \infty }b_{n}\neq 0\end{aligned}}$