Complex Analysis: Theory and Applications · Problem 7.2.9

Taylor and Laurent Series — Howell & Matthews

Mathematics — Complex Analysis · taylor-series

Problem

Let

f(z) = \displaystyle\sum_{n=0}^{\infty} c_n z^n = 1 + z + 2z^2 + 3z^3 + 5z^4 + 8z^5 + 13z^6 + \cdots

, where the coefficients

c_n

are the Fibonacci numbers defined by

c_0 = 1

c_1 = 1

, and

c_n = c_{n-1} + c_{n-2}

, for

n \geq 2

. ((a)) Show that

f(z) = \dfrac{1}{1-z-z^2}

, for all

z \in D_R(0)

for some number

R

. ((b)) Find the value of

R

in part (a) for which the series representation is valid. Hint: Find the singularities of

f(z)

and use Corollary 7.3.