Cauchy products

The Cauchy product is a way to combine two infinite series (like long sums) to create a new series. Imagine each series as an infinite list of numbers. To find their Cauchy product, you multiply each pair of numbers, one from each series, so their positions add up to the same total, then add all these products together. This process produces a new series that captures how the original series interact when combined. It’s a fundamental concept in analysis, especially for studying the behavior and convergence of combined series.