[Resolução] 11.16.21
Posted: 07 Dec 2022 16:17
Prove que:
\(\sqrt{3} = \frac{1732}{1000}(1-\frac{176}{3000000})^{-\frac{1}{2}}\)
Sabendo:
\((1-\frac{176}{3000000})^{-\frac{1}{2}} = (\frac{187489}{187500})^{-\frac{1}{2}} \)
\(\frac{1000 \cdot \sqrt{3}}{1732} = (1-\frac{176}{3000000})^{-\frac{1}{2}} \)
\(\sqrt{3} = \frac{1732}{1000} (1-\frac{176}{3000000})^{-\frac{1}{2}} \)
Sabendo que:
\((1-x)^{-\frac{1}{2}} = \sum_{n=0}^{\infty} \binom{-1/2}{n}(-x)^n\)
Utilizando \(x = \frac{176}{3000000}\)
\( (1-\frac{176}{3000000})^{-\frac{1}{2}} = 1 + \frac{176}{6000000} + \frac{3}{2} \cdot (\frac{176}{6000000})^2 + \frac{5}{2}\cdot (\frac{176}{6000000})^3 + \frac{35}{8}\cdot (\frac{176}{6000000})^4 + \frac{63}{8}\cdot (\frac{176}{6000000})^5 \approx = 1.00003\)
\(\sqrt{3} = \frac{1732}{1000} \cdot 1.00003 \approx 1,7325196 \)
\(\sqrt{3} = \frac{1732}{1000}(1-\frac{176}{3000000})^{-\frac{1}{2}}\)
Sabendo:
\((1-\frac{176}{3000000})^{-\frac{1}{2}} = (\frac{187489}{187500})^{-\frac{1}{2}} \)
\(\frac{1000 \cdot \sqrt{3}}{1732} = (1-\frac{176}{3000000})^{-\frac{1}{2}} \)
\(\sqrt{3} = \frac{1732}{1000} (1-\frac{176}{3000000})^{-\frac{1}{2}} \)
Sabendo que:
\((1-x)^{-\frac{1}{2}} = \sum_{n=0}^{\infty} \binom{-1/2}{n}(-x)^n\)
Utilizando \(x = \frac{176}{3000000}\)
\( (1-\frac{176}{3000000})^{-\frac{1}{2}} = 1 + \frac{176}{6000000} + \frac{3}{2} \cdot (\frac{176}{6000000})^2 + \frac{5}{2}\cdot (\frac{176}{6000000})^3 + \frac{35}{8}\cdot (\frac{176}{6000000})^4 + \frac{63}{8}\cdot (\frac{176}{6000000})^5 \approx = 1.00003\)
\(\sqrt{3} = \frac{1732}{1000} \cdot 1.00003 \approx 1,7325196 \)