Combination Theorem for Sequences/Sum Rule
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Theorem
Real Sequences
Let $\sequence {x_n}$ and $\sequence {y_n}$ be sequences in $\R$.
Let $\sequence {x_n}$ and $\sequence {y_n}$ be convergent to the following limits:
- $\ds \lim_{n \mathop \to \infty} x_n = l$
- $\ds \lim_{n \mathop \to \infty} y_n = m$
Then:
- $\ds \lim_{n \mathop \to \infty} \paren {x_n + y_n} = l + m$
Complex Sequences
Let $\sequence {z_n}$ and $\sequence {w_n}$ be sequences in $\C$.
Let $\sequence {z_n}$ and $\sequence {w_n}$ be convergent to the following limits:
- $\ds \lim_{n \mathop \to \infty} z_n = c$
- $\ds \lim_{n \mathop \to \infty} w_n = d$
Then:
- $\ds \lim_{n \mathop \to \infty} \paren {z_n + w_n} = c + d$
Also see
- Difference Rule for Sequences
- Multiple 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