28437

28437 (Nicolae Mușuroaia)

Fie șirul Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a_n)_{n \geq 1} } cu termenii strict pozitivi, dat de relația $a_{n+1}=ln(a_{1}+a_{2}+...+a_{n}),n\geq 1.$ Determinați $\lim _{n\to \infty }({\frac {a_{n+1}}{a_{n}}}-1)\cdot e^{a_{n}}.$

Soluție:
Pentru orice ${n\geq 2}$ avem $a_{n}=ln(a_{1}+a_{2}+...+a_{n-1})$, deci $a_{n}=a_{1}+a_{2}+...+a_{n-1}=e^{a_{n}}$. Rezultă că pentru orice ${n\geq 2}$ are loc

$$a_{n+1}=ln(e^{a_{n}}+a_{n}).$$

Deoarece $a_{n+1}-a_{n}=ln(e^{a_{n}}+a_{n})-ln(e^{a_{n}}\geq 0)$ pentru orice ${n\geq 2}$ deducem că șirul $(a_{n})_{n\geq 2}$ este strict crescător.
Dacă șirul $(a_{n})_{n\geq 2}$ este mărginit superior, atunci Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a_n)_{n \geq 2} } este convergent cu Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{{n \to \infty}} (a_n) = a \in (0, \infty). } Trecând la limită în relația (1), obținem Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = ln(e^{a_n} + a)} de unde Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = 0 } , absurd! Prin urmare, șirul Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ((a_n)_{n \geq 1}} este crescător și nemărginit superior, deci Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{{n \to \infty}} a_n =\infty} .
Atunci Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{{n \to \infty}}(\frac{a_{n+1}}{a_n}-1) \cdot e^{a_n}=\lim_{{n \to \infty}}\frac{ln(e^{a_n} + a_n)-ln(e^{a_n})}{a_n}\cdot e^{a_n} = \lim_{{n \to \infty}}\frac{ln(1+\frac{a_n}{e^{a_n})}{\frac{a_n}{e^{a_n}}=1 } deoarece din Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{{n \to \infty}} a_n =\infty } rezultă că Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{{n \to \infty}} \frac{a_n}{e^{a_n}}=0}