I don't know why this is true:
$$(2k+1)!!=\frac{(2k+1)!}{2^kk!}$$
Can anyone explain it for me? I know what is double factorial, but would like to know how this formula was derived. Thanks.
[Math] Double factorial formula
factorial
factorial
I don't know why this is true:
$$(2k+1)!!=\frac{(2k+1)!}{2^kk!}$$
Can anyone explain it for me? I know what is double factorial, but would like to know how this formula was derived. Thanks.
Best Answer
HINT : $$(2k+1)!!=1\cdot 3\cdot 5\cdot \cdots (2k-1)\cdot (2k+1)$$ $$(2k+1)!=1\cdot 2\cdot 3\cdot \cdots (2k)\cdot (2k+1) $$ $$2^k\cdot k!=2^k\times \{1\cdot 2\cdot 3\cdot \cdots (k-1)\cdot k\}=2\cdot 4\cdot 6\cdot\cdots (2k-2)\cdot (2k).$$