Let f(z) = Σajzi be the Maclaurin expansion of a function ƒ (z) analytic at the origin. Prove each of the following statements. (a) Σ a¡z²j is the Maclaurin expansion of g(z) := = f (z²). (b) Σajczi is the Maclaurin expansion of h(z) := f(cz). (c) Σ a¡zm+j is the Maclaurin expansion of H (z) := z™ ƒ (z). (d) Laj (z (d) Σaj i=0 (zzo) is the Taylor expansion of G(z) := f (z - zo) around zo.

6. find (a) the minterm and maxterm expansions for F in algebraic and decimal forms, and (b) the minterm and maxterm expansions for F'in decimal form. F(A, B, C) = A + B'C'+ BC.

Consider the function f(t) = 0.01e2t for 0 ≤t≤ π). (a) Sketch the even extension, feven, and the odd extension fodd of the function f(t) over the range -37 ≤ t ≤ 37, clearly indicating each case. (b) Now consider the function g(t) = feven (t) + fodd (t), which has Fourier series G(t) given by 8 G(t) = Ao +) An cos(n7t)+ Σ Αn cos(nat) + Σ Bn sin(nat). B₂ n=1 n=1 Calculate the constant term Ao.

Define f: J -> J by f(1) = 1, f(2) = 2, f(3) = 3, and f(n) = f(n - 1) + f(n - 2) + f(n - 3) for n ≥ 4. Prove that f(n) ≤ 2n for all n in J