The Comparison Test method is a technique used to determine whether an infinite series converges or diverges by comparing it to another known series that has known convergence behavior.
The Comparison Test works by comparing the terms of the given series to those of a known series, typically one that is easier to evaluate. If the terms of the given series are smaller than the terms of the known series and the known series converges, then the given series also converges. If the terms of the given series are larger than the terms of the known series and the known series diverges, then the given series also diverges.
The key concept to understand when using the Comparison Test is that the convergence or divergence of a series is determined by the behavior of its terms. If the terms decrease or increase rapidly enough, the series will converge or diverge, respectively.
The Limit Comparison Test and the Direct Comparison Test are two variations of the Comparison Test method. The Limit Comparison Test compares the limit of the given series to the limit of the known series, while the Direct Comparison Test compares the terms of the given series to the terms of the known series. Both methods can be used to determine convergence, but the Limit Comparison Test is more powerful and can also determine the value of the limit if it exists.
Some common mistakes to avoid when using the Comparison Test include using the wrong known series for comparison, neglecting to check for the absolute convergence of both series, and assuming that the terms of the given series are always larger or smaller than the terms of the known series without proper justification.