27024

27024 (Gheorghe Szöllösy)

Fie 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 I_n = \int_{0}^{\pi} \frac{\cos nx}{13-12\cos x}\,dx, n\ge0.} Să se calculeze $\lim _{n\to \infty }(I_{0}+I_{1}+I_{2}+\ldots +I_{n}).$

Soluție. Să observăm că

$$I_{n+2}+I_{n}=\int _{0}^{\pi }{\frac {2\cos x-\cos(n_{1})x}{13-12\cos x}}\,dx={\frac {1}{6}}\int _{0}^{\pi }{\frac {(12\cos x-13+13)\cos(n+1)x}{13-12\cos x}}\,dx=$$

$$={\frac {1}{6(n+1)}}\sin(n+1)x{\Biggr |}_{0}^{\pi }+{\frac {13}{6}}I_{n+1},$$

oricare ar fi 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\in \mathbb{N}.} 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 I_n=\alpha\left(\frac{2}{3}\right)^n+\beta\left(\frac{3}{2}\right)^n} , 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 \alpha+\beta=I_0=\frac{\pi}{5}} și 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 \frac{2}{3}\alpha+\frac{3}{2}\beta=I_1=\frac{2\pi}{15}.}

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 \alpha=\frac{\pi}{5}, \beta=0} și 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 I_n=\frac{\pi}{5}\left(\frac{2}{3}\right)^n} .

În consecință, 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}(I_0+I_1+I_2+\ldots+I_n)= \frac{\frac{\pi}{5}}{1-\frac{2}{3}}=\frac{3\pi}{5}} .