Comparison Test
Last updated
Last updated
You can understand
Comparison Testintuitively as aSandwich Test.
THIS TEST IS GOOD FOR RATIONAL EXPRESSIONS.
Direct Comparison TestAssume that we have a series a_n, and we're to make up a similar series to it as b_n:
The logic is:
If b > a & b converges, then a converges as well.
If a > b & b diverges, then a diverges as well.
It's so much easier if you think it graphically.
Solve:
Assume the asked series as a_n, and we make up a very similar series bigger than it, as b_n:
Apply the p-series test we get that the b_n converges.
Since a_n < b_n, so according to the Sandwich test (Direct comparison test), a_n converges as well.
Limit Comparison TestLimit comparison test is like an extension when the Direct comparison test won't work. etc., when we compare a with b, although b converges but a > b, so we can't make any conclusion. And that's where the limit comparison test comes in place.
The logic is:
Take the limit of the division a/b.
If the Limit > 0, then they both converges or both diverges.
If the Limit ≤ 0, then there's no conclusion.
We can't easily tell in the comparison who's greater, so we decide to apply the limit comparison test:
Solve:
Solve:
