Direct Comparison Test vs. Limit Comparison Test

Question

How do I determine when to use a Direct Comparison Test versus a Limit Comparison Test for an infinite series?

Answer 1

The limit comparison test is best when you can find a directly comparable function (It is limited, but easier to implement than the limit comparison test when it works)

An easy on is the infinite sum of 1/(n-a) (with a > 0) for which each term is larger than the infinite sum of 1/n. Since the sum of 1/n diverges, it is clear that the first sum also diverges. The same logic can be applied for 1/(n+a)² in which every term is smaller than 1/n² which converges - therefore, the sum of 1/(n+a)² converges.

The limit test is more powerful as you can find that 1/(x+a) diverges also despite every term being smaller than 1/x (The regular comparison test does not show this).

If b_n is the term in the sum we are interested in and a_n is the term form a test function for which the behavior is known, the test is to look at the limiting behavior of the ratio of the terms. If the limiting behavior is the same, then the functions will both converge or both diverge:

limit as n⇒∞ of b_n/a_n = L where L is a finite, nonzero number for our example,

limit as n⇒ ∞ of (n)/(n+a) = 1 so 1/(n+a) series has same behavior as 1/n series

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