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## Calculus, all content (2017 edition)

### Course: Calculus, all content (2017 edition)>Unit 7

Lesson 9: Comparison tests

# Worked example: limit comparison test

AP.CALC:
LIM‑7 (EU)
,
LIM‑7.A (LO)
,
LIM‑7.A.9 (EK)
To use the limit comparison test for a series S₁, we need to find another series S₂ that is similar in structure (so the infinite limit of S₁/S₂ is finite) and whose convergence is already determined. See a worked example of using the test in this video.

## Want to join the conversation?

• Does it matter which we set as "a" and "b" during the limit comparison test?
• Technically, it doesn't matter, because the only requirements for the limit is that it is finite and positive (0 < c < infinity). Because the exact value is not required for this test, it does not matter which is 'a' and which is 'b' for this type of problem.
• Why is it that, even though (2/3)^n is smaller than (2^n)/(3^n)-1, Sal still used it for the comparison? I would have understood if it diverged, because a>b in that case, but since it is converging, isn't b>a supposedly?
• That concept is applied to the direct comparison test. For the limit comparison test, it does not matter if one series is greater than or less than the other series; as long as the limit of their ratios approach a positive finite value, then they are either both convergent or both divergent.
• I really do feel like these videos are fabulous for anyone in Freshman Calculus wanting to further develop their basic intuition a little bit.
• Don't all the functions meet the criteria specified? I realize that the chosen one corresponds the behavior better, but that's rather abstract. But surely the other functions are also zero or greater?
• The limits of the other functions would not equal the limit of the given function and thus they would not be an effective comparison. Thus, the quotient of the limits would not equal a constant between 0 and infinity and the limit comparison test would no longer apply. It therefore could not be stated that both functions either converge of diverge.
Always pick the function that best models the given function. Hope this helps.
• Is there a way to find out what the original S converges to?
• The behaviour of the infinite terms of a series is enough to know whether a series converges or diverges ??
(1 vote)
• No, not in general.

But ∑(2∕3)^𝑛 is a geometric series with a common ratio of 2∕3,
which is less than 1, and thereby the series converges.

By the comparison test we can then conclude that
∑2^𝑛∕(3^𝑛 − 1) also converges.
• At to , Sal says that series a sub(n) and b sub(n) are greater than or equal to zero. Just to clarify b sub(n) cannot be equal to zero it is only greater than zero since it's the denominator.
(1 vote)
• There's nothing saying that 𝑏(𝑛) can not be equal to 0 for some value/s of 𝑛, because we only care about the limit of 𝑎(𝑛)∕𝑏(𝑛) as 𝑛 approaches infinity, and as the video clearly shows, this limit can exist even if 𝑏(𝑛) tends to 0.