How to tell that you've ticked off an engineer

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The check is written for an amount of 0.002 + e^iπ + Σ 1/2ⁿ, as n goes from 1 to ∞.

How much is that?

Well, e^iπ = -1. (See here and here.)

The sum of 1/2ⁿ as n goes from 1 to infinity is equal to 1. That is:
1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ... + 1/∞ = 1

So, the amount of the check is 0.002 + -1 + 1 = 0.002 dollars. That's two-tenths of a cent.