Question 1. Calculate the coefficients Bn in when (10...

Question 1. Calculate the coefficients $B_n$ in when (10 points each) 1) $S(x) = (x + \pi)(\pi - x)$. 2) $S(x) = \sin(x)$. 3) $S(x) = \cos(x/2)$. 4) $S(x) = (1 - x/\pi)$. 5) $S(x) = x/\pi$. $S(x) = \sum_{n} B_n T_n(x)$,

Transcript

00:02 Hi, this series is given here.

00:04 S and x and x, s, the machine, k equal 0 to n and so on.

00:08 Let's evaluate this values here.

00:10 Before, to evaluate these values here, s5, pi over 2, s7, pover, and so on.

00:15 You're to evaluate the series from k equal to 0 to n.

00:19 We'll start here from k equal to 0 and proceeded with this.

00:24 So we can write this series as equals we have, we can just take it as s.

00:32 So as equals we can write it as n p0, sine 0 x, cost, nx, plus nc1, sinex, cost, n, negative 1x, and so on.

00:58 Also, we can write this as equals, we have nc n, sine, nx, cost, 0x.

01:07 We're starting from last term now plus nc and negative 1 we have sign and negative 1 x 4x and so.

01:17 So if we just add here this two series we get 2 s equals we get nc 0, x x x x x plus so for the first term of second series that will be nc nc n x sign 0 x x x x x false 0x then we add second term there'll be nc1 sine x false n negative 1x plus nc and negative 1 1 sign and negative 1 x forth xx next is nc2 sine 2 x2 x plus n negative 2 x plus nc negative 2 x plus nc and negative 2 sine n a 2x cost 2x so we added here the first term to first term second to second term third term and so on so we get 2 as equals let's see here nc 0 nc and are same factor out then so we get sign a cost b plus c plus b that'll be sine a plus b we get fine n x similarly here we have sine a cause b plus or sign b that will give sign nx sign nx so we will have nc 0 plus nc1 plus nc2 and so on till nc nc so we get here nc 0 plus nc1 and so on till nc n and sign an x effect row that's remember 2 to the power n sine x so we get s nx s nx equals 2 to the power n x x equals 2 to the power n negative 1, sine n8.

03:32 Now, the first we evaluate here as 5, pi over 2, that will be 2 to the power 5 negative 1, sign n is 5 times 5 over 2.

03:53 That's coming out to be 2 to the power 4, sine 5 5 5 over 2.

03:59 That is given as sine 2 pi plus pi over 2.

04:06 Now we know that sign pi over 2 is given as 1.

04:14 That will be 16.

04:17 Next we evaluate as 7, pi over 2.

04:27 So perhaps we get 2 to the part n negative 1.

04:33 So 2 to the part 7 negative 1...

Question

Question 1. Calculate the coefficients $B_n$ in when (10 points each) 1) $S(x) = (x + \pi)(\pi - x)$. 2) $S(x) = \sin(x)$. 3) $S(x) = \cos(x/2)$. 4) $S(x) = (1 - x/\pi)$. 5) $S(x) = x/\pi$. $S(x) = \sum_{n} B_n T_n(x)$,

Please give Ace some feedback