Double Integration - Sum approximation - Peter Saulson article on terrestrial gravitational wave antenna

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I'm currently reading an article on terrestrial gravitational noise done by Peter Saulson. He approximates a sum (Eq.4) by an integral with a inner cutoff radius of lambda/4 (Eq.5).

I don't manage to find the same result myself.

I attached the equations from the article + my integral derivation (see attached images)

Does someone know how to obtain the same result as in the paper ?

Regards,

QCP

Equations from the article:
Equations from the article

Self derived integral:
Self derived integral