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In this assignment you will practice solving problems on series, continuity, differentiation & Taylor series. You will also apply these concepts on problems of bond pricing and portfolio hedging. 1. Miscellaneous: (a) Does the following series converge or diverge? n Sn = cos ( TT ) k: 1 1000 , n =1,2, k: k=1 (b) Consider a continuous function, f : R R. Using the definition of continuity, show that the function g (x) = f(3x) is also continuous. (c) Consider the function f(x) = x2 sin (x-1), , x 0, 0, x = 0. Is f differentiable at x = 0, i.e., does f'(0) exist? 2. Power series: Consider the power series 8 F (2)=Enz" = n= 1 (a) What is the radius of convergence of this series? (b) It can be shown that within the radius of convergence, the derivative of a power series can be calculated by term-wise differentiation, i.e., for a power series G(z) = Enzo cnzn, it is the case that (1(2)= = Use this property to derive a simpler expression for F (z), and to calculate F (2/.

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