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.*

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