Convergence theorems are fundamental results in mathematics, particularly in the fields of analysis and probability. They provide conditions under which a sequence of functions or random variables converges to a limit. Common examples include the Dominated Convergence Theorem and the Central Limit Theorem, which help in understanding the behavior of sequences in various contexts. These theorems are essential for establishing the validity of interchanging limits, integrals, and sums. They ensure that under certain conditions, one can confidently analyze the limiting behavior of functions or distributions, making them crucial tools in both theoretical and applied mathematics.