[tex]\displaystyle (\frac{1}{n!}+\frac{1}{(n+2)!})(n+1)!=(\frac{1}{n!}+\frac{1}{(n+2)(n+1)*n!})(n+1)!=\\\\\frac{(n+2)(n+1)+1}{(n+2)(n+1)*n!}(n+1)!=\frac{n^2+n+2n+2+1}{n!(n+1)(n+2)}(n+1)!=\\\\\frac{n^2+3n+3}{(n+2)!}(n+1)!=\frac{n^2+3n+3}{(n+2)(n+1)}(n+1)!=\frac{n^2+3n+3}{n+2}[/tex]