 Hi, I'm Zor. Welcome to Unisor Education. I would like to talk about series, introduction to series. Well obviously this lecture is a follow-up after I was talking about sequences and there are a couple of problems which I actually presented. So what is a series? Well shortly series is a sum of sequence. Usually when we are considering a sequence we have elements which can be actually described as the first, the second, the third, etc. This is element number K and maybe some others. So if it's a finite sequence there is some kind of a maximum order number. If it's an infinite sequence then it's defined for all natural indices. These are indices 1, 2, 3, 4, etc. Example, a sequence of all odd numbers. Now, so series is a sum of a sequence. Usually we are summing up together all the elements of the sequence beginning from the first one, which is A1 or in our example that's the first odd number which is 1 and ending at some maximum number which we are including into summation. So for instance a series which we can basically call as K is a sum of all elements of the base sequence with indices from 1 to K and that's why I use the index K here. Now obviously if the initial, the base sequence is finite and it has let's say n elements then we can construct as 1 as 2, this is A1, this is A1 plus A2, etc. We can construct as n which is sum of all elements up to the maximum available number and no more. Now if it's an internet sequence where there is no maximum number and obviously we can construct series for any index K from 1 to basically to infinity. Well, let's concentrate on the example which I just presented. So let's say we have a sequence of all odd numbers and let's see what are the corresponding series will be. Well, S1 is obviously 1 which is only the first element. S2 is the sum of the first two elements which is 4. Now S3 is the sum of the first three elements which is 9. S4 which is sum of the first four elements is 16, etc. Well, just out of curiosity I hope you notice that this is 1 square, this is 2 square, this is 3 square, this is 4 square and my very intelligent guess is that Sk which is the sum of all first k odd elements would be equal to k square. Well, yes this is not only very intelligent but also correct guess and I'm going to prove it when I will talk about the details of summation and different series. This is just an example and this example is just an illustration of what really a series in this particular case is. This is a series of the sequence of all odd numbers which by the way is a arithmetic progression because it has a difference too and that was my first example. My second example which I would like to present to you is a very ancient one. It was a Greek philosopher by the name Xena and there is a famous Xena's paradox where he is logically proving that Achilles would never be able to catch the tortoise. So let's assume you have the road, this is your tortoise and this is your Achilles which is behind and they are moving this direction. Now let's say initial distance between them is 100 meters. Now let's consider the first interval of time during which Achilles will move 100 meters forward to basically be at the point where tortoise started moving. Now during that same time he has moved to this point tortoise will move a little farther, let's say by one meter forward. Obviously tortoise is a slow-moving object and Achilles is running very fast, he's a Greek hero etc etc so at the time he moves by 100 meters, tortoise moves only a meter. But what I would like to point out that during this time while he is moving to this point and tortoise to this point obviously he didn't catch tortoise. Alright let's consider next period of time. The period of time during which Achilles moves from this point to the point where the tortoise was before. Now he will move one extra meter but she would also move tortoises here, I use sheep. So the tortoise would move some other smaller distance obviously. Let's say it's one centimeters, one centimeter. So by the time he moves one extra meter she moves one extra centimeter so he didn't catch it again. Now obviously this trousers can be repeated infinite number of steps. So it looks like during all these infinite number of steps Achilles would not catch the tortoise. Well obviously we all know that he would sooner or later. So how is this logic actually working? I mean what's the contradiction in this particular logic? Well also when I will talk about geometric progression and the series of the geometric progression I will explain the details of this calculation and I will explain during which period of time or during what distance the Achilles will not be able to catch and then after that he would. So basically I will defeat this particular paradox. I will explain what's wrong with this and logically prove what exactly is the right state of affairs in this particular case. But what's interesting is that this is a perfect example of summation of geometric progression and I'll explain you why. So this is just a couple of examples. One is the sum of odd numbers, sum of arithmetic progression and sum of geometric progression derived from paradox described by Greek philosopher Zina. And the last thing which I wanted to talk about is basically some kind of symbology which is used in mathematics when dealing with sequences and and serieses. Usually to express that the particular, okay, so let's say we have a particular sequence where Achilles is a generalized element of this sequence. For instance, if I'm talking about all odd numbers then generalized odd number will be this. For k is equal to 1, it's 1, for k is equal to 2, it's 3, for k equals 3, it's 5, etc. etc. Now if I want to symbolically explain that I'm summing up all the elements from a1 to an, I will use a Greek letter, uppercase letter sigma and I will put the general member general element of this particular sequence near it. And the only thing which I would like to specify here how exactly I'm summing up, what exactly are the elements which I am summing. So this particular index should be changed from the value of 1 to n because I'm summing up all elements from 1 to n. So that's basically, this is exactly the same as this. And obviously you can use different indices. You can put a i, i from 1 to n or whatever else. I mean this is not important. Letters are whatever you choose but what's important is to use the uppercase Greek letter sigma and specify the limits. Now in many cases, by the way, when we're talking about summing a sequence, arithmetic progression or geometric progression or anything else, we might do a little bit shorter. We might do this. If there is no confusion what's the index, then basically we don't have to specify it here. If we are talking about sequence, usually we are starting from the sequence element number 1. So if we are starting from 1, then only difference basically is what's the maximum number which we are summing up. But anyway, these are just symbols. What I wanted to point out is the usage of the sigma sign for summation. All right. So this is a very short introductory lecture about what actually series is all about. I will spend more time explaining what kind of rules of summation of arithmetic progression or geometric progression or whatever else and I will present a certain number of problems but that's for the future lecture. This is just an introduction. Don't forget to, after you listen to all the theory, try to solve the problems yourself first and only then listen to the lecture which presents some solutions to these problems. And after the lecture, try to do it again yourself just to be more or less to inculcate it in your brain. That's it. Thank you very much for this particular lecture and let's hope the new ones will be more interesting and longer.