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هي مجموعة أرقام مُرتبة بترتيب معين، ونرمز للأرقام بـ a1 و a2 و a3 الى an حيث :

 : الحد الأول ــ  الحد الثانى

 وبشكل عام ،  هو الحد النونى أيا كانت قيمه  .

The sequence {a1, a2, a3, . . .} is also denoted by  an

Definition A sequence \(\left\{a_{n}\right\}\) has the limit \(L\) and we write

\(\lim _{n \rightarrow \infty} a_{n}=L \) or \(\quad a_{n} \rightarrow L\) as \(n \rightarrow \infty\)

if we can make the terms \(a_{n}\) as close to \(L\) as we like by taking \(n\) sufficiently large.

If \(\lim_{n \rightarrow \infty} a_{n}\) exists, we say the sequence converges (or is convergent). Otherwisewe say the sequence diverges (or is divergent).

نظرية هامة جدا:

If \(\lim _{x \rightarrow \infty} f(x)=L\) and \(f(n)=a_{n}\) when \(n\) is an integer, then \(\lim _{n \rightarrow \infty} a_{n}=L\)

Limit Laws for Sequences

If \(\left\{a_{n}\right\}\) and \(\left\{b_{n}\right\}\) are convergent sequences and \(c\) is a constant, then

\(\lim _{n \rightarrow \infty}\left(a_{n}+b_{n}\right)=\lim _{n \rightarrow \infty} a_{n}+\lim _{n \rightarrow \infty} b_{n}\)

\(\lim _{n \rightarrow \infty}\left(a_{n}-b_{n}\right)=\lim _{n \rightarrow \infty} a_{n}-\lim _{n \rightarrow \infty} b_{n}\)

\(\lim _{n \rightarrow \infty} c a_{n}=c \lim _{n \rightarrow \infty} a_{n} \quad \lim _{n \rightarrow \infty} c=c\)

\(\lim _{n \rightarrow \infty}\left(a_{n} b_{n}\right)=\lim _{n \rightarrow \infty} a_{n} \cdot \lim _{n \rightarrow \infty} b_{n}\)

\(\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}=\frac{\lim _{n \rightarrow \infty} a_{n}}{\lim _{n \rightarrow \infty} b_{n}}\) if \(\lim _{n \rightarrow \infty} b_{n} \neq 0\)

\(\lim _{n \rightarrow \infty} a_{n}^{p}=\left[\lim _{n \rightarrow \infty} a_{n}\right]^{p}\) if \(p>0\) and \(a_{n}>0\)

Squeeze Theorem for Sequences

If \(a_{n} \leq b_{n} \leq c_{n}\) for \(n \geq n_{0}\) and \(\lim _{n \rightarrow \infty} a_{n}=\lim _{n \rightarrow \infty} c_{n}=L,\) then \(\lim _{n \rightarrow \infty} b_{n}=L\)

Theorem  If \(\lim _{n \rightarrow \infty}\left|a_{n}\right|=0,\) then \(\lim _{n \rightarrow \infty} a_{n}=0\)


تعريفات هامة جدا:

The sequence \(\left\{r^{n}\right\}\) is convergent if \(-1<r \leq 1\) and divergent for all other

values of \(r .\)

\(\lim _{n \rightarrow \infty} r^{n}=\left\{\begin{array}{ll}{0} & {\text { if }-1<r<1} \\ {1} & {\text { if } r=1}\end{array}\right.\)

Definition A sequence \(\left\{a_{n}\right\}\) is called increasing if \(a_{n}<a_{n+1}\) for all \(n \geq 1\)

that is, \(a_{1}<a_{2}<a_{3}<\cdots\) It is called decreasing if \(a_{n}>a_{n+1}\) for all \(n \geq 1\)

A sequence is monotonic if it is either increasing or decreasing.

Definition A sequence \(\left\{a_{n}\right\}\) is bounded above if there is a number \(M\)

such that

\(a_{n} \leq M \quad\) for all \(n \geq 1\)

It is bounded below if there is a number \(m\) such that

\(m \leq a_{n} \quad\) for all \(n \geq 1\)

If it is bounded above and below, then \(\left\{a_{n}\right\}\) is a bounded sequence.

Monotonic Sequence Theorem Every bounded, monotonic sequence is


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