Sequences

134. Sequences

𝑎_{1}, 𝑎_{2}, 𝑎_{3}, \dots

𝑆 = {𝑎_{𝑖}} 𝑖 \in ℕ = {1, 2, 3, \dots}

Formal definition:

𝑓 : ℕ \to 𝑆

𝑓 (𝑖) = 𝑎_{𝑖}

Convergence

lim_{𝑖 \to \infty} 𝑎_{𝑖} = 𝑎

\forall 𝜀 > 0 \exists 𝑁 \in ℕ \forall 𝑛 \geq 𝑁 : | 𝑎_{𝑛} - 𝑎 | < 𝜀

For every $𝜀 > 0$ , there exists $𝑁 \in ℕ$ such that for all $𝑛 \geq 𝑁$ , $| 𝑎_{𝑛} - 𝑎 | < 𝜀$

Combination of Sequences

𝑔 : continuous \Rightarrow 𝑔 (𝑎_{𝑖}) \to 𝑔 (𝑎)

When does a sequence converge?

𝑎_{𝑖} \leq 𝑎_{𝑖 + 1} \forall 𝑖

Then eiher:

$| 𝑎_{𝑖} - 𝑎 | \leq 𝑏_{𝑖} \forall 𝑖$

and

$𝑏_{𝑖} \to 0$

then

$𝑎_{𝑖} \to 𝑎$