is an increasing sequence
To see this, rewrite and divide two consecutive terms of the sequence:
Since , we have
hence the sequence is increasing. ∎
The sequence is decreasing.
As before, rewrite and divide two consecutive terms of the sequence:
Writing as and applying the inequality , we obtain
hence the sequence is decreasing.
For all positive integers and , we have .
We consider three cases.
Suppose that . Since , we have and . Hence, .
Suppose that . By the previous case, . By theorem 1, . Combining, .
Suppose that . By the first case, By theorem 2, . Combining, . ∎
|Title||is an increasing sequence|
|Date of creation||2013-03-22 15:48:39|
|Last modified on||2013-03-22 15:48:39|
|Last modified by||rspuzio (6075)|