Σ[k=1〜n] (k^2+1)k!=Σ ((k+2)(k+1)-(k+1)-2(k+1)+2)k!
=Σ (k+2)!-(k+1)! -2(k+1)! +2k!
=(n+2)!-2 -2(n+1)!+2
=n(n+1)!

Σ[n=0〜∞] n^4/n!=Σ n(n-1)(n-2)(n-3)+6n(n-1)(n-2)+7n(n-1)+n /n!
=Σ 1/(n-4)! +6/(n-3)! +7/(n-2)! +1/(n-1)!
=(1+6+7+1)e=15e