Mathematics

Problem

Let

k

be a positive real constant and consider the power series

Q_k(x)=\sum_{n=0}^{\infty} \frac{k^{n}}{n!}\,(x+1)^{n}.

Answer the questions below. a. Express the radius of convergence

R(k)

in terms of

k

. Show your reasoning. b. For

k=3

, determine the interval of convergence of

Q_{3}(x)

, including a careful examination of the endpoints. c. For a fixed value

x=2

, find all positive values of

k

for which

Q_k(2)

converges, providing a complete justification.