Prove that the sequence – log n is convergert . [And the constant is denoted by and it is still unknown wheather it is irrational or not ].
Solution :Monotony: It is easy to prove that the sequence is decreasing. Indeed:
The inequality is valid since is a concave function hence lies beneath the line that is its tangent to its graph at . Plugging yields the result.
Boundness: Using the fact that:
we have that . Hence is bounded from below.
Therefore the sequence converges.
The limit of this sequence is a very well known constant, namely Euler – Mascheroni constant.