Chapter 6.2, Problem 64E

Show that fewer than $e\cdot n!$ algebraic operations (additions and multiplications) are required to compute the determinant of an $n\times n$ matrix by Laplace expansion. Hint: Let $L_n$ be the number of operations required tocompute the determinant of a “general” $n\times n$ matrix by Laplace expansion. Find a formula expressing L,, in terms of $L_{n-1}$ . Use this formula to show, by induction(see Appendix B.1), that
$\frac{L_n}{n!}=1+1+\frac{1}{2!}+\frac{1}{3!}+\cdots +\frac{1}{\left(n-1\right)!}-\frac{1}{n!}$ .
Use the Taylor series of $e^x,e^x={\displaystyle \underset{}{\overset{}{\sum }}{}_{n=0}^{\infty }\frac{x^n}{n!}}$ , show that the right-hand side of this equation is less than e.