 Hello friends, welcome to the session of Expansion of Functions Part 3. This is Swati Nikam, Assistant Professor, Department of Humanities and Sciences, Balchan Institute of Technology, Sulakor. At the end of this session, students can create the expansion of functions about any point with the help of standard series, which is obtained by using Maclaurian series. Friends, you can remember all these power series as the first one, exponential series e raised to x contains the terms of ascending powers of x, while the sign series contains terms in ascending odd powers of x with alternate minus and plus sign. In cosine series, it is expanded in ascending even powers of x with alternate minus and plus sign. One more thing you can observe here, in all the three series, each term contains a denominator, which is with the factorial notation and this power and the number present in denominator is same. I have already told you, we are going to solve few examples in this video with the help of these standard series. Example number one, expand f of x is equal to e raised to x into sin x in ascending powers of x up to x cube. Answer, we know that series expansion of e raised to x and sin x is. Now, in previous slide, we have seen expansion of e raised to x and sin x, the same I have written over here. Therefore, product e raised to x into sin x is equal to on RHS in first bracket, I have written e raised to x in the expanded form multiplied by in the second bracket, we have the value of sin x in the expanded form. Therefore, it is equal to, now we have to take product of two brackets term by term. One into second bracket plus x into second bracket plus x square by two factorial into third bracket plus dash dash dash. Now friends, remember that in question, we need expansion up to x cube. So, you can leave the term of higher degree from four and that's why I am not going to consider the other higher degree terms. And instead of that, we have written dash dash shine here. So, it is equal to one into first bracket. So, I am going to consider the terms up to x cube only. So, there are two terms x minus x cube upon three factorial and for the rest term plus dash dash dash plus. Now, see here from this second product, it is x into x that is x square, leave all other higher degree terms. So, dash dash dash plus from the third product, it is x square into x that is x cube upon two factorial and for all other terms dash dash dash dash. Therefore, e raised to x into sine x is equal to x plus x square plus in bracket one by two minus one by six into x cube plus dash dash dash. Which is equal to x plus x square plus two by six into x cube plus and so on. And finally, the simplified form for e raised to x into sine x is equal to x plus x square plus one by three into x cube plus dash dash dash. Which is the required answer. Now friends, before we proceed for the next example, write the expansion of first a plus b whole square, second a plus b whole cube and third a minus b whole cube. I hope you have written your answer. So, by using binomial theorem, the expansion of a plus b whole square is a square plus two a b plus b square. Second, a plus b whole cube is equal to a cube plus three a square b plus three a b square plus b cube. And third one, a minus b whole cube is equal to a cube minus three a square b plus three a b square minus b cube. Let us solve second example. Prove that cos cube x is equal to one minus three by two x square plus seven by eight x raised to four dash dash dash. Answer, we know that cos x is equal to series expansion is one minus x square by two factorial plus x raised to four by four factorial minus x raised to six by six factorial plus x raised to eight by eight factorial dash dash dash. Therefore, cos cube x is equal to, so on RHS in bracket, we will write down series expansion of cos x and then whole cube, which is equal to, let us write down the first term one minus bracket all other terms in second bracket and then whole raised to three, which is in the form of first term minus second term whole cube. Now, we are going to use one result a minus b whole cube, which is equal to a cube minus three a square b plus three a b square minus b cube, where one is nothing but a and this bracket is nothing but b. So, we have done such type of arrangement. Therefore, cos cube x is equal to first term one cube minus three into first term square into second term as it is plus three into first term into second term square minus second term cube. So, it is equal to one as it is minus three into x square by two factorial, this minus minus plus three times x raised to four upon four factorial. So, dear friends, again remember that here in expansion of proof, we need the terms up to x raised to four. So, for higher power terms, you can put dash dash sign and hence we are going to consider the product terms here up to x raised to four for rest terms put dash dash dash sign. While writing the terms from third term or third product, we are going to use result a minus b whole square. So, for that a is the first term minus b is all other rest term. But when you write down a minus b whole square, it is first term square. So, which is x square by two factorial whole square and all other terms are of higher degree than four. That's why we are going to leave them and keep a dash dash dash sign. Same is the thing with the last expansion. So, if you want to expand it, it is a minus b whole cube which is started with a cube. A cube is the first term cube where first term is x square by two factorial whole cube and which is obviously of higher degree. So, leave it. This is equal to one minus three by two into x square plus three into x raised to four divided by four factorial is 24 plus three into x raised to four divided by two factorial square that is two square is four plus dash dash dash dash. So, after simplifying our course cube x becomes one minus three by two whole square plus seven by eight x raised to four plus and so on. I have referred a text of applied mathematics by NP Bali for creation of this video. Thank you so much for watching this video and have a happy learning.