Limit of a Summation #1

$\begin{aligned} \frac{i^2 + i - 1}{(i-2)! + (i-1)! + i! + (i+1)!} &= \frac{i^2 + i - 1}{(i-2)!\left[ 1+\left( i-1 \right) +\left( i-1 \right) \left( i \right) +\left( i-1 \right) \left( i \right) \left( i+1 \right) \right] }\\ &=\frac{i^2 + i - 1}{(i-2)!\left[ 1+i-1+{ i }^{ 2 }-i+{ i }^{ 3 }-{ i }^{ 2 }+{ i }^{ 2 }-1 \right] }\\&=\frac{i^2 + i - 1}{(i-2)!\left[ { i }^{ 3 }+{ i }^{ 2 }-i \right] }\\&=\frac{1}{(i-2)!i}\\&=\frac{i-1}{i!}\\&=\frac{1}{(i-1)!}-\frac{1}{i!}\\\lim_{n \to \infty} \displaystyle \sum_{i=2}^{n} \frac{i^2 + i - 1}{(i-2)! + (i-1)! + i! + (i+1)!} &=\lim_{n \to \infty} \displaystyle \sum_{i=2}^{n} \frac{1}{(i-1)!}-\frac{1}{i!}\\&=\frac { 1 }{ 1! } -\frac { 1 }{ 2! } +\frac { 1 }{ 2! } -\frac { 1 }{ 3! } +\frac { 1 }{ 3! } -\frac { 1 }{ 4! } +... \\&\Huge\color{#3D99F6}{=\boxed{1}}\end{aligned}$