A convergent series is like a race. Imagine a hundred runners competing in a race. Everyone starts running and eventually, one runner will cross the finish line first. The other runners will eventually make it across the finish line too, but in a different order and at different speeds.
That's kind of like a convergent series - it's a series of numbers that start out with different values, and then get closer and closer together until they all have the same value in the end. This means that the series converges, or ends up at a single value.