27020

27020 (Gheorghe Szöllösy)

Să se calculeze suma $\sum _{k=0}^{\left[{\frac {n}{2}}\right]}{\frac {1}{4^{k}\cdot (k!)^{2}\cdot (n-2k)!}},\quad n\geq 1$

Soluție:

Fie $a_{n}$ coeficientul lui $X^{n}$ din rezolvarea lui

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 P(X) = \left(X + \frac{1}{2}\right)^{2n} = \left(X(1+X) + \lfloor 1/4 \rfloor\right)^n = \sum_{k=0}^n C_n^k (! - X)^{(n-k)} \left(\frac{1}{4^k}\right)}

Avem 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 = \left(\frac{1}{2^n}\right) C_2n^n } , iar pe de altă parte, 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 = C_n^0 \cdot C_n^0 + C_n^1 \cdot C_(n-1)^1 \left.\frac{1}{4}\right. + C_n^2 \cdot C_(n-2)^1\left.\frac{1}{4^2}\right. + ... = } 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 = \sum_{k=0}^{\left[\frac{n}{2}\right]} C_n^k C_(n-k)^k \cdot \left.\frac{1}{4^k}\right. = n! \sum_{k=0}^{\left[\frac{n}{2}\right]} \frac{1}{(k!)^2 (n-k)! 4^k},}

deci suma este egală 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 \left.\frac{(2n!}{2^n(n!)^3}\right. .}