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 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 = a_1 + a_2 + ... + a_{n-1} = e^{a_n}} . Rezultă că pentru orice 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 {n \geq 2} } are loc
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+1}=ln(e^{a_n} + a_n).}
Deoarece 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+1} - a_n = ln(e^{a_n} + a_n) - ln (e^{a_n} \ge 0) } pentru orice 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 {n \geq 2}} deducem că 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 2} } este strict crescător.
Dacă 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 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