Let .
- Prove that is strictly increasing on .
- If , solve the equation
Solution
- It is . However, recalling that
we conclude that is strictly increasing.
- We note that
On the other hand, and hence:
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Let .
Solution
we conclude that is strictly increasing.
On the other hand, and hence:
Evaluate the product
Solution
We are making use of the identity
Hence,
Let be a strictly increasing sequence of positive integers. Prove that the series , where denotes the least common multiple, converges.
Solution
We have successively:
Let . We define the sequence . Prove that
Prove that
Solution
Let and . Hence,