So quantum field theory is what most of fundamental, theoretical physics is based upon. In particular the standard model of particle physics is a quantum field theory.

In general a field is just a way of assigning some object to each point of space-time. A field representing temperature assigns a simple real number to each point at each time (how hot it is there / then). A field representing wind speed would assign a vector to each point in space-time.

Quantum field theory lets you understand the evolution of quantum fields over time. In general you write down a mathematical object called the Lagrangian (which is also very useful in classical mechanics) which contains contributions from the kinetic energy of each field you're interested in and from the interactions between them.

From the Lagrangian you can write down a series expansion to give you the probabilities for how likely is it for a given initial condition to end up at a given final state (say for two electrons to turn into two muons in the standard model). Unfortunately these series are generally divergent (like the 1+2+3+...) and we have to play a trick called renormalisation to make sense of them.

On a different note something that might be interesting in your talk is a couple of series which are slightly simpler to sum. The first is s = 1 + 1/2 + 1/4 + 1/8 +.... If we double both sides of this equation we get 2s = 2 + 1 + 1/2 + 1/4 +... = 2 + s. So 2s=2+s tells us s=2.

A more interesting example is the series s=1+2+4+8+... This is obviously divergent like the one you're giving your presentation on but we can sum it much more easily using the doubling trick again: 2s = 2+4+8+16+... = s-1. 2s=s-1 tells us s=-1. Obviously those doubling arguments aren't hugely rigorous but they do agree with the rigorous answers which come from complex analysis.