Why limit comparison test is inconclusive if L=0 or L=infinity, calculus 2 tutorial

Learn why we cannot draw any conclusion if L=0 or L=infinity when we use the limit comparison test. Remember, for the limit comparison, we check the limit of an/bn. And if the limit is finite and greater than 0, then we can say either the series of an and series of bn both converge or both diverge. But check out the series of sin^2(1/n). If we compare this with the series of 1/n, then we will get L=0 (or L=infinity if we compute limit bn/an instead). If we want to say that the series of sin^2(1/n) and series of 1/n both converge, then we will be drawing the wrong conclusion.

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#justcalculus

[gentlemandude]

This is a very good demonstration video of why the limit comparison test is inconclusive in particular situations, but it doesn't really explain *why* this is the case, which makes the title a bit misleading. That being said, it is a very good illustration of the scenarios that can affect the interpretation of the results from the limit comparison test.

[Avighna]

The realization that you had to use \sum 1/n^2 instead of \sum 1/n was so smart!

[GoldrushGaming]

I never really knew why, I just accepted it as true. Its nice to know why things happen in math

[Josh-tfcr]

Therapist: "Shaolin monk blackpenredpen doesn't exist, he can't hurt you."
Shaolin monk bprp:

[conrad]

I am wondering about the cases L=0 or infinity. Comparing to a converging series b_n, why can't I conclude with L=0 a_n is converging?

[alexandregeraldes]

hello and thanks for everything.
I have a question !!

How to find my p

to complete this test

in this case you chose p=2
how to know which p to choose??

[yigitcem]

Hi sometimes when we switch the denominator we get both 0 and infinity for example an=sin(1:n)/n bn=1/n we get 0, that means we cant get any conculusion about sin(1/n)/n, but if we switch them we get infinity then if the L=infinity we conclude that sin(1/n)/n diverges. And thats wrong, What am I missing at this point.

[arimermelstein]

My tl;dr version would be that if a_n and b_n grow at the same rate, then your limit will be in R^+. It’s the same reason that when you want to find if one function grows at the same rate using O notation, you can define it as the limit of the quotient tending to a constant.

[alimousavi]

This must be pinned if you love your fans.