Step 1 :\(R_1 = \lim\limits_{n \to +\infty} \frac{n!(n+1)^2}{(n+1)!(n+2)^2} = \lim\limits_{n \to +\infty} \frac{n+1}{n+2} = 1\)
Step 2 :\(R_2 = \lim\limits_{n \to +\infty} \frac{(3+2i)^n(n+1)^2}{(3+2i)^{n+1}(n+2)^2} = \lim\limits_{n \to +\infty} \frac{n+1}{(3+2i)n+2(3+2i)} = \frac{1}{3+2i}\)
Step 3 :\(\Delta U(x, y) = U_{xx}(x, y) + U_{yy}(x, y) = e^x + 2 + (-2) = e^x\)