The purpose of using comparison tests and limit comparison test is to determine the convergence or divergence of a series or sequence. These tests help to compare the given series or sequence with known series that have already been proven to converge or diverge, making it easier to determine the behavior of the given series.
Comparison tests and limit comparison test work by comparing the given series or sequence with a known series that has a similar behavior. If the known series converges, then the given series is likely to converge as well. If the known series diverges, then the given series is likely to diverge as well. This comparison allows for a more efficient and accurate determination of convergence or divergence.
The main difference between comparison tests and limit comparison test is that comparison tests compare the given series with a known series term by term, while limit comparison test compares the ratio of the terms in the given series with the ratio of the terms in a known series. Additionally, comparison tests are used for series with positive terms, while limit comparison test can be used for series with both positive and negative terms.
Comparison tests and limit comparison test should be used when trying to determine the convergence or divergence of a series or sequence. They are especially useful when the series or sequence is complex and it is difficult to determine its behavior through other methods. These tests can also be used to check the accuracy of other convergence or divergence tests.
No, comparison tests and limit comparison test cannot be used for all series and sequences. They are most effective for series with positive terms and for series that are not too complex. Additionally, limit comparison test may not work for series with alternating signs or series with terms that do not approach zero as n approaches infinity.