Exercitii limite

Lauryca Mar Oct 30, 2012 12:03 am

Am 2 probleme de rezolvat la limite,care nu le dau de cap.
1.Se considera sirurile a(n) si b(n) care au aceeasi limita l. Sa se arate ca sirul c(n), c(n)=(a(n)+b(n))/2 + ((-1)^n)/2 (a(n)-b(n)) are limita l.
2.Sirul a(n) are termenii a(2n)<0 si a(2n-1)>0 pentru oricare n apartine N*.
a) Poate fi convergent acest sir?
b) Poate avea acest sir limita +infinit? Dar -infinit?

**Admin** Mar Oct 30, 2012 1:39 am

1.Se considera sirurile a(n) si b(n) care au aceeasi limita l. Sa se arate ca sirul c(n), c(n)=(a(n)+b(n))/2 + ((-1)^n)/2 (a(n)-b(n)) are limita l.

$\begin{array}{l} c\left( n \right) = \frac{{a\left( n \right) + b\left( n \right)}}{2} + \frac{{{{\left( { - 1} \right)}^n}}}{2}\left[ {a\left( n \right) - b\left( n \right)} \right]\\ n = {\rm{par}} \Rightarrow c\left( n \right) = \frac{{a\left( n \right) + b\left( n \right) + a\left( n \right) - b\left( n \right)}}{2} = \frac{{2a\left( n \right)}}{2} = a\left( n \right) \to l \Rightarrow c\left( n \right) \to l\\ n = {\rm{impar}} \Rightarrow c\left( n \right) = \frac{{a\left( n \right) + b\left( n \right) - a\left( n \right) + b\left( n \right)}}{2} = \frac{{2b\left( n \right)}}{2} = b\left( n \right) \to l \Rightarrow c\left( n \right) \to l \end{array}$

Exercitii limite

Exercitii limite

Re: Exercitii limite