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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2016 • Question 6

Question

Math 6420 • PDE Fall 2016 • Question 6

The following function \(f(x)\) [\(x\) is a real variable] can be represented by a series (Taylor or Laurent) in powers of \((x-7)\). Find the radius of convergence of the series in each case \[ \begin {array}{rcl} \text {(a)}\quad f(x)&=&e^{(x-7)^{10}},\\[0.6ex] \text {(b)}\quad f(x)&=&\left (\dfrac {\sin x-3}{\sin x-2}\right )^2,\\[1.4ex] \text {(c)}\quad f(x)&=&\dfrac {\sin x-2}{x^2-49}. \end {array} \] [You do not need to find the series themselves.]

Raw LaTeX

The following function \(f(x)\) [\(x\) is a real variable] can be represented by a series
(Taylor or Laurent) in powers of \((x-7)\). Find the radius of convergence of the series in
each case
\[
\begin{array}{rcl}
\text{(a)}\quad f(x)&=&e^{(x-7)^{10}},\\[0.6ex]
\text{(b)}\quad f(x)&=&\left(\dfrac{\sin x-3}{\sin x-2}\right)^2,\\[1.4ex]
\text{(c)}\quad f(x)&=&\dfrac{\sin x-2}{x^2-49}.
\end{array}
\]
[You do not need to find the series themselves.]

Topic Tags

Consistency, Stability, and Convergence Forward Euler and First-Order Convergence Laurent Series and Annuli Linear Programming and Optimization