Siruri de numere reale Modalitati de definire a sirurilor

Astazi o sa discutam despre siruri de numere reale, adica o sa studiem monotonia sirurilor dar si marginirea sirurilor.
Dupa cum bine stiti exista mai multe modalitati de definire a sirurilor, dar mai intai sa definim notiunea de sir.
Definitie: O functie $f:N^{*}\rightarrow R$ se numeste sir de numere reale si se noteaza $\left(a_{n}\right)_{n\geq 1}$
Modalitati de definire ale sirurilor
– Siruri definite cu ajutorul unor formule

$a_{n}=\frac{3n+1}{n^{2}}$
-siruri definite printr-o relatie de recurenta
$a_{1}=1, a_{n+1}=\frac{1}{2}a_{n}+\frac{3}{2}$ , oricare ar fi $n\in N^{*}$
–siruri definite descriptiv
Incepem printr-un exemplu simplu pentru a studia monotonia sirurilor dar si marginirea sirurilor
1) Fie sirul $a_{n}$ dat de relatia
$a_{n}=\frac{n^{2}+2n}{\left(n+1\right)^{2}}$
a) Sa se studieze marginirea sirului
b) Sa se studieze monotonia sirului
Solutie
a) Stim ca
$a_{n}=\frac{n^{2}+2n}{\left(n+1\right)^{2}}=1-\frac{1}{\left(n+1\right)^{2}}<1\Rightarrow a_{n}\in \left(0,1\right)$
Deci sirul este marginit.
Acum studiem monotonia sirului, cel mai simplu ar fi sa calculam:
$a_{n+1}-a_{n}=\frac{\left(n+1\right)^{2}+2\left(n+1\right)}{\left(n+1+1\right)^{2}}-\frac{n^{2}+2n}{\left(n+1\right)^{2}}=\frac{n^{2}+2n+1+2n+2}{\left(n+2\right)^{2}}-\frac{n^{2}+2n}{\left(n+1\right)^{2}}= \frac{n^{2}+4n+3}{\left(n+2\right)^{2}}-\frac{n^{2}+2n}{\left(n+2\right)^{2}}=\frac{\left(n^{2}+4n+3\right)\cdot\left(n+1\right)^{2}-\left(n^{2}+2n\right)\cdot\left(n+2\right)^{2}}{\left(n+1\right)^{2}\left(n+2\right)^{2}}=\frac{\left(n^{2}+4n+3\right)\cdot\left(n^{2}+2n+1\right)-\left(n^{2}+2n\right)\cdot\left(n^{2}+4n+4\right)}{\left(n+2\right)^{2}\cdot\left(n+1\right)^{2}}=\\ \frac{n^{4}+2n^{3}+n^{2}+4n^{3}+8n^{2}+4n+3n^{2}+6n+3-n^{4}-4n^{3}-4n^{2}-2n^{3}-8n^{2}-8n}{\left(n+2\right)^{2}\cdot\left(n+1\right)^{2}}=\frac{2n+3}{\left(n+2\right)^{2}\cdot\left(n+1\right)^{2}}>0$
deci sirul este crescator
2) Fie sirul $a_{n}=\frac{n^{2}+1}{n^{2}+2}$ sa se studieze:
a) marginirea sirului
b) monotonia sirului
Solutie
$a_{n}=\frac{n^{2}+1}{n^{2}+2}=1-\frac{1}{n^{2}+2}<1\Rightarrow a_{n}\in \left(0, 1\right)$ .
Deci sirul este marginit.
b) Monotonia, calculam:
$a_{n+1}-a_{n}=\frac{\left(n+1\right)^{2}+1}{\left(n+1\right)^{2}+2}-\frac{n^{2}+1}{n^{2}+2}=\frac{n^{2}+2n+1+1}{n^{2}+2n+1+2}-\frac{n^{2}+1}{n^{2}+2}=\frac{n^{2}+2n+2}{n^{2}+2n+3}-\frac{n^{2}+1}{n^{2}+2}=\frac{\left(n^{2}+2n+2\right)\cdot\left(n^{2}+2\right)-\left(n^{2}+1\right)\cdot \left(n^{2}+2n+3\right)}{\left(n^{2}+2n+3\right)\cdot\left(n^{2}+2\right)}=\frac{n^{4}+2n^{2}+2n^{3}+4n+2n^{2}+4-n^{4}-2n^{3}-3n^{2}-n^{2}-2n-3}{\left(n^{2}+2n+3\right)\cdot\left(n^{2}+2\right)}=\frac{2n+1}{\left(n^{2}+2n+3\right)\cdot\left(n^{2}+2\right)}>0\Rightarrow a_{n}$ crescator.

Sau monotonia o putem arata si cu definitia pentru siruri monotone, astfel calculam

$a_{1}=\frac{1^{2}+1}{1^{2}+2}=\frac{2}{3}=0,6$

$a_{2}=\frac{2^{2}+1}{2^{2}+2}=\frac{4+1}{4+2}=\frac{5}{6}=0,83$

$a_{3}=\frac{3^{2}+1}{3^{2}+2}=\frac{9+1}{9+2}=\frac{10}{11}=0,90$

$a_{4}=\frac{4^{2}+1}{4^{2}+2}=\frac{16+1}{16+2}=\frac{17}{18}=0,94$

Deci abservam ca

$a_{1}<a_{2}<a_{3}<a_{4}$ , deci sirul este monoton crescator.

Siruri marginite

Definitie: Spunem ca sirul $\left(a_{n}\right)_{n\in N}$ este marginit daca exista un interval $\left[a,b\right]$ care contine toti termenii sirului $a\leq a_{n}\leq b, \forall n\in N$ .

Observatie. Definita de mai sus este echivalenta cu existenta unui numar real M>0 astfel incat $|a_{n}|<M, \forall n\in N^{*}$ .

Sirurile care nu sunt marginite se numesc nemarginite.

Monotonia sirurilor

Sirul $\left(a_{n}\right)_{n\geq 1}$ se numeste sir monoton crescator daca $a_{n}\leq a_{n+1}, \forall n\in N^{*}$ , adica $a_{1}\leq a_{2}\leq a_{3}\leq ...\leq a_{n}\leq...$ .

Sirul $\left(a_{n}\right)_{n\geq 1}$ se numeste sir monoton descrescator daca $a_{n}\geq a_{n+1}, \forall n\in N^{*}$ , adica $a_{1}\geq a_{2}\geq a_{3}\geq ...\geq a_{n}\geq...$ .

Observatie. Daca inegalitatile de mai sus devin stricte, adica $a_{n}<a_{n+1}$ sau $a_{n}>a_{n+1}$ , atunci sirul se numeste strict crescator sau strict descrescator.