Answer Happy

Problem L Power series are typically used to "break” a function into a sequence of numbers (the Taylor coeffi- cients of the function). However, sometimes it is useful to go in the opposite direction, assembling a sequence of numbers into a function. Let fn be the n-th Fibonacci number of Example 3c in 8.1, k=n k=n An = k=1+2+...+n, Bn = $k = 12 + 22 + ... + n°; = k=1 k=1 by definition fo = Ao = Bo= 0. (a) Give a recursive definition of the numbers fn, An, Bn with n> Q (b) Use mathematical induction and only part (a) to show that fn, An, Br <5" for all n> Q (c) Use the Absolute Convergence and Comparison Tests and only part (b) to show that the power series f(x) = { fn. 2e", = A(z) = È Anxo", = B(x) = Ž Bmx", = n=0 n=0 n=0 converge if || < 1/6 (and thus determine smooth functions near x=0). (d) Using only part (a), show that 2 f(x) = x+xf(x) +x+ f(x), A(x) = xA(x) + (1-x)2' B(x) = xB(2) + (1-2) 2:02 + (1-x)3 Hint: You'll need to use identities such as the following: I/ 00 1 1 (n »-:(-)- (83")*- Enir – 1}<h2, (Σ" -1) 1 1 - 2)3 1 1 21-2 n=0 n= (e) Using only part (d), express fn, An, and Bn explicitly in terms of n. Hint: use (d) to solve for f, A, and B and expand them into Taylor series around x=0 (partial fractions might help in the case of f); compare the result with the definitions of f, A, and B in (c) Note: For fn, you should end up with the formula in Problem G on PS6. There is a much simpler way of finding an explicit formula for An; so you can check your answer, but please deduce this formula from (b). The answer for Bn can be confirmed using induction (or google).