1. Taylor
2. Application of power series
3. Fourier Series

1. Taylor series
   • We found that if represent a function as a power series
     

     f(x) = a0+a1(x-a)+...+an(x-a)n+...,

     that
     

     an = f(n)(a) n! .

   • Maclaurin series are Taylor series where a = 0.
   • If f(x) = 1/x then want f⁽n)(a) to find taylor series generated by this. x^-1 goes to (-1)x^-2, which goes to 2!x^-3. So nth derivative is n!(-1)ⁿ x^-(n+1). This means that
     

     an = (-1)n n! x-(n+1) n! = (-1)nx-(n+1).

     So if expand about x = 2 then a = 2 so our taylor series is
     

     å n  (-1)n (x)-(n+1)(x-2)n = 1 2 - (x-2)2 22 + (x-2)2 23 +¼.

   • This is a geometric series with r = -(x-2)/2 and first term 1/2. Converges for r<1 or | x-2 | < 2. Know what the infinite sum is for this:
     

     S = a 1-2 = 1/2 1+(x-2)/2 = 1 2+x-2 = 1 x .

   • Taylor Polynomials of order n, P_n(x), are series where we just keep the first n terms. They provide the best polynomical approximation of the relevant function possible using polynomials of nth degree!
   • When can we expect taylor series to converge to the function we are approximating and how accurate is our approximation?
     • function is equal to P_n(x)+R_n(x) where R_n(x) is the remainder. It's absolute value is called the error associated with the approximation.
     • Taylor's theorem:If f is differentiable through order n+1 in an open interval I containing a then for each x in I there exists a number c between x and a such that
       

       Rn(x) = f(n+1)(c) (n+1)! (x-1)(n+1).

       This is a generalisation of the Mean value theorem. If R_n(x)® 0 as n® ¥ then we say that the series generated by f at a converges to f on I.
     • Estimating the Remainder:If there are positive constants M and r such that | f⁽ⁿ⁺¹⁾(t) | £ Mrⁿ⁺¹ for all t between x and a, inclusive then the remainer term satisfied the inequality
       

       | Rn(a) | £ M rn+1 | x-a | n+1 (n+1)! .

       Normally what we say the error is O(hⁿ⁺¹) but the derivative term need not be small which is why we need results like this. In general the derivative being large doesn't turn out to be a problem.

   • If have common interval of convergence then can add, subtract and multiply taylor series term-by-term.

2. Application of power series
   • Binomial series:Use for (1+x)³ = x³+3x²+3x+1. With taylor can deal with (1+x)^1/2 by expanding about 0. Then
     

     = 1+ x 2 -[1/2(-1/2)]1/2! x2+¼.

   • Note that this only converges for x < 1 since if radius greater than one would include 1/0 when the function is no longer continuous. Also do not want to include negative numbers in our interval of convergence since the square root of negative numbers are not real number!
   • Series solutions of differential equations.
   • If have
     

     dy dx -y = x

     with y(0) = 0, as in an initial value problem then we can represent y as a series as well as y'. This lets us solve the equation.
     

     y¢ = a1+2a2x+....+nanxn-1+¼

     so
     

     y¢-y = (a1-a0)+(2a2-a1)x+...+(nan-an-1)xn-1+¼ = x.

     So we have 2a₂-a₁ = 1 and a₁ = a₀, a₀ = 1 since y(0) = 1 as well as na_n-a_n-1 = 0 for the other cases. So a₀ = 1 and a₁ = a₀ = 1. Then 2a₂-a₁ = 1 so a₂ = 1 = 2/2, but 3a₃-a₂ = 0 so a₃ = 1/3, a₄ = a₃/4 and so on. Can write as a_n = 2/n! for n ³ 2.
     

     y = 1+x+x2+x2+2x3/3!+...

     which is the same as
     

     y = 1+x+2(x2/2!+x3/3!+...) = 1+x+2(ex-1-x) = 2ex-1-x.

3. Fourier series
   • Express functions as
     

     f(x) = a0 2 + å n  (an cos npx L + bn sin npx L ).

     Need to determine a_n and b_n. This can easily be done since
     

     ó õ L -L  cos npx L cos mpx L dx = L

     if n = m or else its zero. Same for sin. sin cos is always zero. So
     

     an = 1 L ó õ L -L  f(x) cos np L dx.

     and
     

     bn = 1 L ó õ L -L  f(x) sin np L dx.

     
     

     a0 = 1 L ó õ L -L  f(x)dx.

4. Won't be asked on fourier or application of power series to DE's which you will see a lot of next year and beyond.
5. Will bring in sample papers on tuesday and put up solutions for the homework problems on the web during the week. Remember to ask question in tutorials.

---

File translated from T_EX by T_TH, version 2.70.
On 3 May 2002, 13:15.