27024

27024 (Gheorghe Szöllösy)

Fie $I_{n}=\int _{0}^{\pi }{\frac {\cos nx}{13-12\cos x}}\,dx,n\geq 0.$ 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=

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

oricare ar fi $n\in \mathbb {N} .$ AtunciFailed 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} șiFailed 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,

\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}}.