Very nice diagrammatic explanation! I always get a kick out of the use of that geometric series to get the improved propagator. You are right to have the minus sign before the self-energy term, which I missed in my propagator expression.
Following @SamuelGWalters post about Feynman propagators.
Note that fermionic propagators are matrices, unlike bosonic ones
We define a 1PI diagram as a diagram which does not fall into two pieces if you cut one internal linepic.twitter.com/odVPmRehik
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