Probleme siruri convergente

George95 Lun Dec 17, 2012 11:59 pm

Buna seara!Ma ajutati va rog la rezolvarea unui exercitiu?
Se considera sirurile:
$a_{n}=n-\sum_{k=1}^{n}\frac{k^{2}+3k+1}{k^{2}+3k+2}$
$b_{n}=a_{1}a_{2}...a_{n},n\geq 1$

**Admin** Mar Dec 18, 2012 2:45 am

$\begin{array}{l} \sum\limits_{k = 1}^n {\frac{{{k^2} + 3k + 1}}{{{k^2} + 3k + 2}}} = \sum\limits_{k = 1}^n {\frac{{{k^2} + 3k + 2}}{{{k^2} + 3k + 2}} - \sum\limits_{k = 1}^n {\frac{1}{{{k^2} + 3k + 2}}} } = n - \sum\limits_{k = 1}^n {\frac{1}{{{k^2} + 3k + 2}}} \\ \\ \frac{1}{{{k^2} + 3k + 2}} = \frac{1}{{k + 1}} - \frac{1}{{k + 2}} \Rightarrow \sum\limits_{k = 1}^n {\frac{1}{{{k^2} + 3k + 2}}} = \sum\limits_{k = 1}^n {\frac{1}{{k + 1}}} - \sum\limits_{k = 1}^n {\frac{1}{{k + 2}}} = ... = \frac{n}{{2n + 4}}\\ \\ {a_n} = n - n + \frac{n}{{2n + 4}} = \frac{n}{{2n + 4}} \to \boxed {{\color{Red} \frac{1}{2}}} \end{array}$

Natura b_n ... tema!

Probleme siruri convergente

Probleme siruri convergente

Re: Probleme siruri convergente