The (Direct) Comparison Test

[@narutokurosaki6477]

my teacher soent an entire class period teaching this and I had no idea what he was even talking about. I see 5 minutes of this video, and I understood it completely. thanks so much

[@Mathispower4u]

Thank you for the comments!

[@rodolfobarranco03]

just a quick question, in minute 4 you compare the function 1/(n-2)^(1/2) to 1/n^(1/2) which is actually smaller than the original function. since the definition of the comparison test says that in order for this to work the function An has to be between 0 and Bn. in other words: 0

[@alkajon]

When writing out a factorial, you are multiplying by the starting number and every integer down to 1. For example 3! = 3 * 2 * 1. So if you have n!, as n increases, so does the factors that you are multiplying by, whereas the base of 2 you only ever multiply by 2 for any value of n-1. 3! > 2^2 because 6 > 4 and 4! > 2^3 because 24 > 8. Essentially, the factorial increase faster than the exponent.

[@SarkTheShark94]

bullcleo1 > patrickJMT I find your videos more organized and clear. Thanks Man

[@shaheenhearts]

you just took something that looks so horrible and scary on paper and turned it into something human friendly!! THANK YOUU SO MUCH!!!

[@ethanlarue4881]

@ 5:13 why are we saying is it >? and not

[@williamwarren9397]

how do I know if its inconclusive?

[@willytillyfilly]

OMG THANK YOU FOR UPLOADING!!!!! You're a lifesaver. :D

[@3andhalfpac]

you also could use 1/n^2 starting n at 4 since 4^2 is 16 and 4! is 24 as long as 1/n!> 1/n^2.

[@DADDY557]

i think that is relating this n! to the"growth rates of sequences" which states that n! grows faster than b^n ....... that's how i think he got that random thing

[@nycityzen]

Why did you choose to compare n! to 2^n?

[@nuggets5787]

I think it's because 2^n is a geometric series. Factorial and geometric series look similar that's why he compared them together. Just a thought, just wanted to shear.

[@liangpingshen9910]

When you expand n!, you will get n^n-something, for n>=2, n^n>=2^n

[@nahuel3256]

@@nuggets5787 that it is not the reason.

[@nahuel3256]

The reason is because 2^(n-1) is always smaller or equal than n! Therefore, the reciprocal, that is: 1/2^(n-1) will always be greater or equal than 1/n! In this way, if the series that is "greater" converges, the series that is "smaller" will also converge by the comparison test.

[@sameergowani]

we could also use the root test in the second example right?

[@ac7491]

That is an interesting cursor you have there.

[@Mathispower4u]

Yes, that is correct.

[@muhammadkabir-ud-dinnawaz8706]

Where can I get PDFs of these slides ?

[@exile341]

I don't understand example #2 Bn is smaller than An. The rule for the comparison test states that 0

[@rinaberman7686]

For divergence, the rule is 0

[@rinaberman7686]

People like to leave that part out for some reason.

[@TheTechnicalPandyaji]

what is the straterg to know which test to apply when

[@aoides]

why 2^n is use to compare with n!? thanks

[@jasonjkeller79]

@ 6:03 I'm a little unclear as to how you know to use 2^n as a comparison to n!. I understand the following steps but using 2^n doesn't seem immediately obvious to me. Any tips for spotting these things more easily?

[@audiomac]

I completely agree. It's not clear. I think it's not expected to be clear for us, but I think we are supposed to memorize and remember that pattern.

[@ΔημήτρηςΠπαπαδόπουλος]

great work !!! very helpful !!!

[@MrVumile]

2^n is not clear for me....think u should explain more on it.

[@kaicool71]

but first we have to check that Un/Vn is a finite quantity then only we can apply the comparison test

[@maithaap4565]

thank you , such a life savor

[@namansaxena8999]

Thanks man Really helpful. ..

[@yash1133]

could we have used the integral test here.

[@kadenkks]

If you can do an integral, the integral test is always possible. Some integrals are difficult or stupidly hard compared to other tests, which is when comparisons and ratio tests come in handy

[@yash1133]

kadenkks thnks

[@moeabangir8171]

great video thank you

[@syedmahbubulhuda6762]

Very helpful

[@milano1400]

so fucking helpful cheers man!!!! :D

[@abby4899]

How in the world did you come up with that last one to compare the factorial question to? Did you just make it up? Please DO NOT ASSUME that students know! Write out the algebra because that's where the marks are. If the algebra is wrong then everything is wrong so please show all the working. I really don't know why most teachers loooove assuming that students know. UGH

[@witsleep]

+Psychedelix he did not assume, he did do the a little bit of explanation (not that it help) but in exam, you either find it out yourself what the statement does or just know it and factorial is just one case, so it's not that bad

[@crazyfool333]

i fucking love you right now!

[@liangpingshen9910]

When comparing 1/(n^2+3) and 1/n^2 , you could do it in a simpler way: because n^2+3>n^2, 1/(n^2+3)

[@MrAlbaraa123]

thank you

[@daveburton8332]

1

[@HandicappedNoobsCA]

THANKKK YOUUUU SOOOOO MUCHHHHHHHH

[@ronchertakovsky2872]

Why did you choose to compare n! to 2^n?

[@HARCONSTRUCTION]

Because n= power and n!=(n)!