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.
(a) Does the following series converge or diverge?
Sn = cos ( TT ) k:
(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
x2 sin (x-1), , x 0,
x = 0.
Is f differentiable at x = 0, i.e., does f'(0) exist?
2. Power series: Consider the power series
F (2)=Enz" =
(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
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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