nwr sorted this out above. There are several mathematical models that have infinite time and an infinite chain of cause and effect. All of classical mechanics and classical statistical mechanics is like this. So infinite time cannot be "logically" ruled out.
Also, again more for others than arguing with WookieeB, there isn't just one Logic. Research into logic since the 1930s has discovered that there are entire families of alternate logics with different rules of inference for combining propositions.
People have known how to sum infinite series in special cases since the 3rd century BC in Greece, developed even more in India in the Middle Ages and fully developed between the 17th-19th centuries. Today it's taught from around the age of 15 in many national curricula. Hardly some unknowable mystery.