LaTex mathematical symbols + code running results

\[\Gamma_{ij}^{k}=\frac{1}{2}(\frac{\partial g_{il}}{\partial u^j}+\frac{\partial g_{jl}}{\partial u^i}-\frac{\partial g_{ij}}{\partial u^l})\]\[\sum_{i,j,k=1}^{\infty}{x_{i_{j_{k}}}}\]\[\frac{f(x)}{g(x)}\]\[\frac{x}{y}+\frac{f(x)}{g(x)}\]\[\lim_{x\rightarrow x^0}f(x)=A\]\[\lim_{x\rightarrow x_0}f(x)=A\]$\int^a_dc_b f(x)dx$\(\int^a_dc_b f(x)dx\)\[\int^a_dc_b f(x)dx\]\[\sum_{i=1}^{\infty} x_i\]\[\sqrt[5]{x^4-3x+1}\]Errors=\(\sqrt{\frac{\sum_{j=1}^{M}[predictior(j)-real(j)]^2}{M}}\)\[\iint_{\Omega}f(x,y)dxdy\]\[\iiint_{\Gamma}f(x,y,z)dxdydz\]\[\left|\begin{array}{cccc}1 & 6 & 9 \\7 & 90 & f(x)\\9 & \psi(x) & g(x)\end{array}\right|\]\[\left[\begin{array}{cccc}1 & 6 & 9 \\7 & 90 & f(x)\\9 & \psi(x) & g(x)\end{array}\right]\]\[\left(\begin{array}{cccc}1 & 6 & 9 \\7 & 90 & f(x)\\9 & \psi(x) & g(x)\end{array}\right)\]\[\left(\begin{array}{llll}1 & 6 & 9 \\7 & 90 & f(x)\\9 & \psi(x) & g(x)\end{array}\right)\]\[\left(\begin{array}{rrrrr}1 & 6 & 9 \\7 & 90 & f(x)\\9 & \psi(x) & g(x)\end{array}\right)\]\[\begin{cases}\ u_{tt}(x,t)= b(t)\triangle u(x,t-4)&\\\ \hspace{42pt}- q(x,t)f[u(x,t-3)]+te^{-t}\sin^2 x, & t \neq t_k; \\\ u(x,t_k^+) - u(x,t_k^-) = c_k u(x,t_k), & k=1,2,3\ldots ;\\\ u_{t}(x,t_k^+) - u_{t}(x,t_k^-) =c_k u_{t}(x,t_k), &k=1,2,3\ldots\ .\end{cases}\]\[q(x,t)=\begin{cases}(t-k+1)x^2,\quad \ \ &t\in\big(k-1,k-\dfrac{1}{2}\big],\\(k-t)x^2, \quad \ \ & t\in\big(k-\dfrac{1}{2},k\big],\end{cases}\]

Code running result: