Combination Theorem for Sequences/Multiple Rule
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Theorem
Real Sequences
Let $\sequence {x_n}$ be a sequences in $\R$.
Let $\sequence {x_n}$ be convergent to the following limit:
- $\ds \lim_{n \mathop \to \infty} x_n = l$
Let $\lambda \in \R$.
Then:
- $\ds \lim_{n \mathop \to \infty} \paren {\lambda x_n} = \lambda l$
Complex Sequences
Let $\sequence {z_n}$ be a sequence in $\C$.
Let $\sequence {z_n}$ be convergent to the following limit:
- $\ds \lim_{n \mathop \to \infty} z_n = c$
Let $\lambda \in \C$.
Then:
- $\ds \lim_{n \mathop \to \infty} \paren {\lambda z_n} = \lambda c$
Also see
- Sum Rule for Sequences
- Difference Rule for Sequences
- Product Rule for Sequences
- Quotient Rule for Sequences
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): convergent series
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): convergent series