Convergent Sequence

analysis

Definition

Convergent Sequence

A sequence $(a_n{)}_{n\ge 0}$ is called convergent if it has a limit.

Example: The limit of the below sequence is:

$$\underset{n\to \infty }{\lim }{a}_n=a$$

which can also be written as:

$$a_n⟶a$$

Improperly Convergent

Definition

Uneigentlich Konvergente Folge

Improperly Convergent Sequence

A sequence $(a_n{)}_{n\ge 0}$ whose elements become arbitrarily large with:

$$\forall K>0:\exists N(K)\in \mathbb{N}:\forall n>N(K):a_n>K$$

and can be denoted as as:

$$\underset{n\to \infty }{\lim }{a}_n=\infty$$

Analogously, one defines ${\lim }_{n\to \infty }{a}_n=-\infty$, and such sequence are also called improperly convergent. The values $+\infty ,-\infty$ are then referred to as improper limits.

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Limitedness

All convergent sequences are bounded.

Let $(a_n{)}_{n\ge 0}$ a sequence with ${\lim }_{n\to \infty }{a}_n=a$ and $\epsilon >0$. The sequence elements with $a_n\notin U_{\epsilon }(a)$ are $a_{n_1},a_{n_{n_2}},\dots ,a_{n_k}$. Let $\bar{\epsilon }>{\max }_{i=1,\dots ,k}|a-a_{n_i}|$ (or $\bar{\epsilon }=\epsilon$ in the case $k=0$). Then, in particular, $\bar{\epsilon }\ge \epsilon$, and hence all sequence elements lie in $U_{\epsilon }(a)$. The interval boundaries $a-\bar{\epsilon }$ and $a+\bar{\epsilon }$ then form the lower and upper bounds of $(a_n{)}_{n{\ge }_0}$.

Cauchy Sequence

Definition

Cauchy Sequence

A real sequence is called cauchy sequence, named after Augustin-Louis Cauchy, if for all $\epsilon >0$ there exists an $N(\epsilon )$, such that:

$$\forall n,m>N(\epsilon ):|a_n-a_m|<\epsilon$$

Cauchy sequences are convergent and all convergent sequences are Cauchy sequences.

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