Question

завершите равенство :
n/(n+1)! = ? / n! - ? / (n+1)!

Answer 1

$\frac{n}{(n+1)!}=\frac{A}{n!}-\frac{B}{(n+1)!}\\\\\frac{A}{n!}-\frac{B}{(n+1)!}=\frac{A}{n!}-\frac{B}{n!\, (n+1)}=\frac{A(n+1)-B}{n!\, (n+1)}=\frac{An+A-B}{(n+1)!}\\\\\frac{n}{(n+1)!}=\frac{An+A-B}{(n+1)!}\\\\An+A-B=n\; \; \; \Rightarrow \\\\A\cdot n+(A-B)\cdot n^0=1\cdot n+0\cdot n^0\; \; \; \Rightarrow \\\\n^1\; |\; A=1\; ,\\n^0\; |\; (A-B)=0\; \; \to \; \; \; 1-B=0\; ,\; \; B=1\\\\\frac{n}{(n+1)!}=\frac{1}{n!}-\frac{1}{(n+1)!}\\\\\\\star \; \; (n+1)!=n!\, (n+1)\; \; \star$