15 - Introduction to sequences

Real Sequence

A real sequence or sequence of real numbers is a function
$α : N \to R$
where $α (n)$ is the $n^{th}$ term of the sequence
Then you can also represent the sequence as
$(a_{n})_{n \geq 1} = (a_{n})_{n = 1}^{\infty} = a_{1}, a_{2}, \dots,$

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Complex Sequence

Similar to Real Sequence but with
$a : N \to C and a_{n} \in C for n \geq 1$

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Note that the order of the terms in a sequence are very important!

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Notation on sequences

$(a_{n}) a_{n} represents the sequence represents the term of the sequence$

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Termwise operations on Sequences

Let $(a_{n}), (b_{n})$ be sequences then new sequences can defined termwise e.g.
$(a_{n} + b_{n}), (- a_{n}), (a_{n} b_{n}), (c a_{n}), (∣ a_{n} ∣)$
If $b_{n} \neq = 0$ for all $n$ then we can also define
$(\frac{a _{n}}{b _{n}})$

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Oxford Maths Notes

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15 - Introduction to sequences

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