Use Newton's method to find the first 5 approximations of the solution to the equation $LaTeX: \displaystyle \cos{\left(x \right)}= \frac{123 x^{3}}{1000} - 3$ using $LaTeX: \displaystyle x_0=3$ .

Using the formula for Newton's method gives $LaTeX: x_{n+1} = x_{n} - \frac{- \frac{123 x_{n}^{3}}{1000} + \cos{\left(x_{n} \right)} + 3}{- \frac{369 x_{n}^{2}}{1000} - \sin{\left(x_{n} \right)}}$ Using $LaTeX: \displaystyle x_0 = 3$ and $LaTeX: \displaystyle n = 0,1,2,3,$ and $LaTeX: \displaystyle 4$ gives: $LaTeX: x_{1} = (3.0000000000) - \frac{- \frac{123 (3.0000000000)^{3}}{1000} + \cos{\left((3.0000000000) \right)} + 3}{- \frac{369 (3.0000000000)^{2}}{1000} - \sin{\left((3.0000000000) \right)}} = 2.6213324513$ $LaTeX: x_{2} = (2.6213324513) - \frac{- \frac{123 (2.6213324513)^{3}}{1000} + \cos{\left((2.6213324513) \right)} + 3}{- \frac{369 (2.6213324513)^{2}}{1000} - \sin{\left((2.6213324513) \right)}} = 2.5939015685$ $LaTeX: x_{3} = (2.5939015685) - \frac{- \frac{123 (2.5939015685)^{3}}{1000} + \cos{\left((2.5939015685) \right)} + 3}{- \frac{369 (2.5939015685)^{2}}{1000} - \sin{\left((2.5939015685) \right)}} = 2.5937681993$ $LaTeX: x_{4} = (2.5937681993) - \frac{- \frac{123 (2.5937681993)^{3}}{1000} + \cos{\left((2.5937681993) \right)} + 3}{- \frac{369 (2.5937681993)^{2}}{1000} - \sin{\left((2.5937681993) \right)}} = 2.5937681961$ $LaTeX: x_{5} = (2.5937681961) - \frac{- \frac{123 (2.5937681961)^{3}}{1000} + \cos{\left((2.5937681961) \right)} + 3}{- \frac{369 (2.5937681961)^{2}}{1000} - \sin{\left((2.5937681961) \right)}} = 2.5937681961$