let's go over eight examples where we have to use the ratio test or maybe the root test to help us to see if a series converges or not here's the first one the series as n goes from 1 to Infinity of n/ 3 to the N it does look like a geometry Series right and you can really say this right here is greater than or equal to the series as Ang go from one to Infinity of 1/ 3 to the N but the problem with this is that this right here actually converges because the common ratio is 1 over three and this is greater than because n is greater than or equal to one yeah but the thing is it does not help us to draw any conclusion when you say this right here is greater than a convergent series you are saying this is greater than a finite number well it can be just a bigger finite number or it can be Infinity we cannot draw any conclusion so in this case we can actually try the ratio test why the ratio test though here's the key when you have an in the exponent maybe you want to give it a try well may be the root test but uh the root test you have to take the N root of n not the pretus so ratio test might be better when you have any the exponent what factorial try the ratio test so by racial test we are just going to do the following this right here is our a n so we are going to check the limit as n goes to infinity and remember the ratio test is that it's the absolute value of a n + 1 over a n we can write it as a n + 1 * 1/ a n so the n+ one term plug in plug in and then times the reciprocal of the original so this right here will give us the limmit as n goes to Infinity the absolute value doesn't matter because here everything's positive n is positive a n + one you just put n+1 into here and here so n + one over 3 to the n + 1 times the reciprocal of the original so just do that 3 to the n/ N now check this out 3 to the n + 1 we can break it apart as 3 to the n * 3 to the 1st moreover 3 to the N 3 to the N cancel so we are looking at the limit as n goes to Infinity on the top is just n + 1 on the bottom we have 3 * [Music] n now s n goes to Infinity for this kind limit just compare this and that which is going to tell us the limit is 13 and when we get the limit of this being equal to less than one not equal to if it's equal to one the ratio test can help us draw any conclusion this right here is less than one therefore we can say this right here converges and we know this limit cannot be negative because we appli the absolute value already so right here we can come here and say the series converges by ratio test but do not brief ratio test because root test is also RT so just spell the out by ratio [Music] test now for the second one we see the factorial and we see n in the exponent so definitely try the ratio test so here is our a n here I will just say we are going to check and let me write down what we are going to check again for you guys the limit as n goes to Infinity of the absolute value of a n + 1 * 1/ a n so we see that this is the limit as n goes to Infinity the absolute value doesn't matter because again everything is positive a n + 1 just put M plus1 here and m+1 here so we get two Q to the m + 1 over here this is M +1 that we're entering yeah we also have to enter the M plus1 here but this another plus one and then after that we do the factorial and then the reciprocal of that which we have n + 1 factorial over 2 to the N so this right here it's really just m + 2 factorial and now how do we break this apart well we can write this down as m + 2 and then times the next term which is n+ one and then times the next term which is n and so on so on so on which we can say is n + one factorial always try to break down the bigger factorial and also the bigger exponent likewise here this is 2 to the n + 1 which is 2 to the n * 2 to the 1st now happy canceling 2 to the n 2 to the N cancel and then n +1 factorial n +1 factorial cancel have a look here is the limit as n goes to Infinity on the top we have this two to the first power so it's just two no more and on the bottom is n + 2 as n goes to Infinity we have 2 over infinity so in this case the limit equal to zero and now you have to remember we are doing the ratio test if the limit that we get is less than one this is all we need to say and that tells us it converges so come back here and we say this right here converges by the ratio test [Music] it's the limit comparon test that we cannot say anything if the limit is equal to zero but for the ratio test if the limit is equal to zero we can still draw conclusion as long as the limit is less than one that series converges now for number three series as n goes from 1 to Infinity of 2 n + 1 raised to the n's power over n raised to the 3 n power this time we have both n in the power here and here here so it would be a better idea to use the root test because when we take the N root they pretty much get cancelled yeah you can still try the ratio test too but I will leave that to you guys so here we go we are going to use the root test so what we are going to check is the limit as n goes to infinity and we take the N root and then here just in case sometimes we have negative stuff but don't worry we will take the absolute value take the absolute value and again this right here is our a an and we're just going to put the a n in here so here we go we will do the limit as n goes to infinity and then we take the N root the N root which is the same as the 1 over n power very nice the absolute value doesn't matter in this case because everything is positive so we can just write this down which is 2 m + 1 1 raised to the n's power over n to the 3 n power now because we have this over that so when we do this power to that power n and one over cancel likewise when we do this to that power n and one over cancel so we are looking at the limit n goes to Infinity here it's just 2 n + 1 on the top on the bottom it's just n to the third power on the bottom now as n goes to Infinity you care about this you care about that it reduces to 2 over n 2 and as n goes to Infinity we get zero and for the root test as soon as the limit is less than one we get to draw conclusion that this right here converges by the root test [Music] now for number four the series as Ang goes from 1 to Infinity of -4 rais to the n's power and then here we have 3 N squared firstly you can rewrite this as the series as n goes from 1 to Infinity of -1 to the N yeah because this is like netive 1 * 4 and then here we have 4 to the n's power over 3 to the n Square power you can do that and you can try to do this with the alternating series test but well in this video we are talking about either the ratio or the root here we have n in the power so a better way is the root test here we go we are going to check the limit as n goes to Infinity the N root along with the absolute value and then we put the a n inside I'm just going to write this down for a n so this is the limit as n goes to Infinity the n r is the same as the 1 over n power on outside here and then the top here is -1 to the n and then 4 to the N over 3 to the N SAR here I'm still going to keep the absolute value because I'm actually going to use the absolute value to get rid of this term 1 to the N power is1 + 1 1 + 1 right so when you have the absolute value for that this is just going to be always one so we are just going to get let me just write down one more line to make it super clear so focus on this to the 1/ n's power and then 4 to the N over 3 n squar all right now 4 to the N raised to the 1/ n n and n cancel yeah so this is going to be I'll write this down [Music] here just four on the top and on the bottom this right here is three to the N because you can just divide this by n so n² over n that's how we get the N right [Music] here okay and then as n goes to Infinity the bottom here is three to the infinity which is inity on the top is just four so this right here gives us zero therefore we know it's less than one therefore this right here convergence by the root test now for number five we have the series as angles from one to Infinity of 1 over square root of n factorial so this right here is not P series do not say this is n to the 1/2 power because we have the factorial inside and because we have the factorial try the ratio test so here we go we check the limit as n goes to infinity and then what we do is the absolute value of a n + 1 * 1 / a n so this right here is our a n so now this right here is the limit as n goes to infinity and AB for doesn't matter because this is always positive a n + 1 put in here so we have 1 over square root n + 1 and then we do the factorial times the reciprocal of the original so we get just square root of n factorial over one like that let's put on over one why not now break down the bigger factorial but this time they still inside the square root so we are looking at square root this is n + 1 times the next term which is n times the next term which is nus1 and so on which is just n factorial n +1 factorial equals n +1 * n factorial and this right here is just the same as Square < TK of n + 1 * < TK n factorial and you know this and that can cancel so we get get the limit as n goes to Infinity 1 / squ < TK of n + 1 as n goes to Infinity this limit gives us zero we have less than one therefore this right here converges by the ratio test now for number six the series as angle goes from 1 to Infinity of 3 to the N over n * 2 to the N again this is not a geometric series because we have this extra end multiplying two to the end and you might try to do some comparison test but I'll will leave that to you here let's try to use the ratio test because I really don't want to deal with the N root of n right as n goes to Infinity so ratio test will be the way to go we check the limit as n goes to Infinity absolute value of a n + 1 * 1/ a n all right so I'm going to just put n + one into all the NS so this is the limit as n goes to Infinity we have 3 to the n + 1 over n + 1 * 2 n + 1 and then the the reciprocal of that which is n * 2 to the N over 3 to the N now breaking apart this right here is 3 to the n * 3 to the 1st this right here is the same as 2 to the n * 2 to the 1st 3 to the N 3 to the N cancel and 2 to the n 2 to the N cancel all right this is the limit as n goes to Infinity on the top we have 3 n on the bottom we have two * that so it's 2 * n + 1 so in this case we are actually looking at the limit as n goes to Infinity then me just multiply this in to make it super clear this time as n goes to Infinity as we can see the limit is 3 over 2 and that is greater than one therefore this series Divergence by the way this is a circle to keep my video consistent anyway though this right Divergence like the first one in this video right Divergence by ratio test now number seven the series as n goes from 1 to Infinity of n factorial over n to the n's power because of the factorial it's a great idea to use the ratio test so this is our a n right let me Circle that and then we check the limit as n goes to Infinity of the absolute value a m + 1 * 1 / a n so we get the limit and goes to Infinity ABS value again doesn't matter and we will just have a n + 1 so n + 1 here and then we factorial that over n + 1 raised to the n + 1 and then times the reciprocal of that which is n to the N Over N factorial now for the top it's just n + 1 * n factorial so you know this and that can cancel but for this one a little bit stranger right I can break this part as n + 1 to the n * n + one to the first and then in this case we can happen to cancel n +1 n+1 here but we cannot cancel this and that so be careful with this but now we are looking at the limit as Ang goes to Infinity notice how they both have the n in the exponent so so we can write it down as n over n + 1 and then raised to the n's power again because of this n and n hm how can we deal with this though if n goes to Infinity the inside you get one but n goes to Infinity in fact this is a one to the infinity in determinant form we cannot draw any conclusion right now and the idea is that we can actually use one of the secret weapons that can help us out and let me write that down right here for you guys I call that the fact it's a very useful limit fact it's a limit as n goes to Infinity of 1+ a Over N raised to the bn's power here you get three letters e to the A B's power so I'm going to show you how to put put this into this form first though we are going to flip the fraction inside so we get the limit as n goes to infinity and I really want to have n + one over n like this so I can split the fraction well what we have to do is just negate the exponent the negative exponent allows us to to flip the fraction now we can split the fraction this is the limit as n goes to Infinity of n / n is 1 plus 1 / n and then raised to the negative n's power now this is in this form and we can say that a is just equal to 1 and B is1 therefore the limit for this is e to the 1 * -1 which is -1 and for this one you have to be careful e to the 1 is 1 over e and because we're doing the ratio test you mention this right here is less than one and you come here and draw the conclusion that the series converges by the ratio test and we are [Music] done now for the last one we have this now for the last one the series as Ang goes from one to Infinity of 1 + 2 N over 3 + 4 n and then raised to the n's power don't you just really want to get rid of the n's power I will totally do so too and to do so we use the root test so here we are going to check the limit as n goes to infinity and then we take the N root and then here we have put the absolute value and then the a n this is the a n and then just do it here is the limit as n goes to Infinity the N through is the 1 / n's power and the absolute value doesn't matter because everything is positive I will just put down 1 + 2 N over 3 + 4 n and then rais to the n's power the beauty for this is this is not cancel right away so we are going to get the limit as n goes to infinity and then we have 1 + 2 N over 3 3 + 4N then for this limit we just need to worry about 2 N and 4N reduce you get 1/2 Which is less than one and we can compy and say this right here convergence by the root test so that's it hopefully this video is helpful if you have any questions feel free to leave a comment Down Below in let me know