| (n+1)!(2n)! | ||
dany jest ciąg an = | , n należy do N dodatnich zbadaj monotoniczność | |
| (2n+1)!n! |
| n!(n + 1) * (2n)! | n + 1 | |||
an = | = | |||
| (2n)! * (2n + 1) * n! | 2n + 1 |
| n + 2 | n + 1 | |||
an + 1 − an = | − | = | ||
| 2n + 3 | 2n + 1 |
| (n + 2)(2n + 1) − (n + 1)(2n + 3) | ||
= | = | |
| (2n + 3)(2n + 1) |
| 2n2 + 5n + 2 − 2n2 − 5n − 3 | −1 | |||
= | = | < 0 | ||
| (2n + 3)(2n + 1) | (2n + 3)(2n + 1) |
| (n+2)!(2n+2)! | (n+1)!(2n)! | |||
an+1−an= | − | = | ||
| (2n+3)!(n+1)! | (2n+1)!n! |
| n+2 | n+1 | |||
= | − | |||
| 2n+3 | 2n+1 |
Teraz wspólny mianownik zrób
| −2n−1 | |
<0 | |
| (2n+3)(2n+1) |