Converging integration series

Calculus Level 3

Prove that the series converges:(See the serie carefully) In the options by e it means Euler Masheroni constant

$1 - \int_1^2 \dfrac {dx}x + \dfrac12 - \int_2^3 \dfrac{dx}x + \dfrac13 - \cdots$

Also find the sum of the expression

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