Let denote the Euler – Mascheroni constant. Define . Prove that

**Solution**

Well let ; since increase to on we deduce by the monotone convergence theorem that

Now,

Hence

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Let denote the Euler – Mascheroni constant. Define . Prove that

**Solution**

Well let ; since increase to on we deduce by the monotone convergence theorem that

Now,

Hence

The above identity is due to Catalan (1875). The above also satisfies the relation

(1)

which is a typical example of Mahler’s functional equations in the theory of transcedental numbers. Also note that

(2)