Hello students myself Dr. Gajendra Purohit and today I am going to start a new topic sequence and series and sequence and series is a very important topic. This creates problem for students as this is a topic for advance calculus. In this any sequence that is given OK? for that sequence we make the infinite series and how much is the value of that infinite series and if it comes finite it is convergent and if it comes infinite- plus infinite or minus infinite then it is divergent and by chance if the value does not come to be unique means it is coming finite but it is not unique means the value keeps changing. So this type of sequence is oscillating sequence. Students if here we are given any infinite series and we have to find about that infinite series that if the infinite series is convergent or divergent or oscillating series then we can find that with the help of a test. Here we are going to discuss about many tests in this video. So my first video is the introductory video where the series what is the concept of the sequence? and what is the concept of the series? what is convergent and divergent? what is bounded above and bounded below? OK? what is monotonic increasing and monotonic decreasing sequence. So this is some basic ideas I am going to tell you. The geometric series in that what is the criteria for the series to be convergent. Students I am going to discuss all of this. Students before this lot of my videos are uploaded for engineering mathematics, bsc students where I have uploaded videos for IIT-JEM and even for GATE. So students here on the I icon you can go and watch all my videos. Do like my videos and do subscribe to my channel and press the notification bell so that for the upcoming videos you keep getting the notifications. So students see here what is sequence? First of all we study this topic on sequence in 11th class where we talk about arithmetic progression and geometric progression and that is our sequence. So the meaning of the sequence is that the arrangement of numbers follow some or the other formula. Right? Like I say we have 1,2, 3 so see here what happens 1 is getting added an it is gong till infinity. So we call this sequence but why? because here it is +1 like I take one more example for you here I take 1/2, 1/4 1/8 Right? So this is also a sequence but what type of sequence? So this is a sequence. Right? Now we talk about that if the sequence is convergent or divergent. So students see here, let me tell you one thing that this sequence is the increasing sequence in this+1 is getting added. Clear? So now whenever it will be added it will go till infinite so students the value of this increase so if we do the sum for this then the value will be infinite means it will increase. Right? So this sequence will be divergent sequence. because its value, now divergent means that for the sum we cannot find the value. Right? We cannot find that what will be the sum of the sequence? So if now I talk abut this in this 1/2 is getting added and you can see that if this is 1/2 then the next term is smaller to that. So this is a decreasing sequence. Right? So if this is a decreasing sequence then at one point its value will be 0 so this type of sequence is the decreasing sequence and if we sum it up then the value that comes is a finite value. So students this type of sequence is the convergent sequence. Clear? So here I just gave you a basic idea that what is convergent and divergent? Clear? Now I will talk about that this I told you for a sequence and how what is series? So the sum for these like for any given sequence like this I told you for a very simple one but there can be many more sequence in which a pattern is followed or a formula is followed. So we call it a sequence. Now for that sequence the sum of the infinite series So we find the sum for this and the sum is coming finite or infinite means not finite or infinite So this I talked about that this type of series is known as infinite series but the nth term for this I will tell you all in the definition that what is the concept behind convergent and divergent but I am telling you that the sequence is of this type What is the sum of the nth term whether it is finite or infinite or it is coming finite but not unique so for this we decide the definition that it is convergent, divergent or oscillatory sequence. Clear? So students here the name of the topic is infinite series. So the meaning of the infinite series is that for the sequence the infinite term sum . Clear? So we check for that if it is convergent means the value for this is finite or infinite or it is not unique. Clear? So students the next topic that I am going to explain you now the definitions you can see in the book but the term that is being used that I will tell you. So here the term that will be used that it bounded sequence. So what is bounded sequence students. Here we have two types of terms one is bounded below and the other one is bounded above. So students what is the meaning of bounded below? So it is that sequence for which we know the below term but the upper term Now students you must have a question that why are we discussing this the topic on bounded because the connection of sequence is also with the boundedness. Any sequence that is convergent so it will be 100% bounded. But if some sequence is bounded but if we talk about convergency so it is not necessary for it to be convergent . Like I take an example in front of you I will tell you that what is oscillatory sequence. I was telling all of you that bounded below, bounded above and bounded concept and what it is? Students now I will talk about what is monotonic increasing sequence? and what is decreasing sequence? After that we will talk about convergency that what are convergent, divergent and oscillating sequence? So students here I want to tell you some concept on monotonic increasing sequence and monotonic decreasing sequence and what it is? Students this is the concept of 12th class and you know that what is the concept of increasing and decreasing but then also I will give you little idea Now students what is convergent, divergent and oscillating sequence. So students So students what I taught you is infinite series that for infinite series the sequence the series is convergent, divergent or oscillating series and I even explained you the basic concept for bounded above and bounded below. So students the tests I am going to start now as my next video that will come in which I will you that how we find whether the series here is convergent or divergent. So the next video that I will take that is comparison test and after that the series will follow. So keep following our channel and soon I will complete this topic. Students you must be liking my videos and if you like them then do subscribe to my channel. Do press the notification bell so that for all the upcoming videos