# Casimir force using Pauli-Villars regularization

In Zee's Quantum field theory in a nutshell, 2nd edition, p. 74 he claims that:

$$\sum_a c_a \Lambda_a \sum_n \frac{\omega_n}{\omega_n + \Lambda_a} = - \sum_a c_a \Lambda_a \sum_n \frac{\Lambda_a}{\omega_n + \Lambda_a}$$

can be proved using

$$\sum_a c_a \Lambda_a = 0$$

This doesn't seem to make any sense. Note that $\omega_n$ can be both larger and smaller than $\Lambda_a$. Perhaps I should note that the $\Lambda_a$ are taken to be "large". Also, $\omega_n \propto n$ and the sum over n goes from 1 to $\infty$.

Has anyone gone through this derivation successfully?

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Add and subtract $\Lambda_a$ in the numerator on the left-hand side. The term with the minus is the result on the rhs, while the term with the plus along with the $\omega_n$ cancels because of the stated condition.
To make this explicit: $\omega/(\omega + \Lambda) = 1 - \Lambda/(\omega + \Lambda).$ –  Vibert May 4 '13 at 13:31