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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2015 • Question 6

Question

Math 6420 • PDE Fall 2015 • Question 6

The following functions \(f(x)\) [\(x\) is a real variable] can be represented by series (Taylor or Laurent) in powers of \((x-2)\). Find the radius of convergence of the series in each case \[ \begin {array}{rcl} \text {(a)}\quad f(x)&=&e^{x^2},\\[0.6ex] \text {(b)}\quad f(x)&=&\dfrac {\sin x-3}{\sin x-2},\\[1.2ex] \text {(c)}\quad f(x)&=&\dfrac {e^x}{(x-2)(1+e^x)}. \end {array} \]

Raw LaTeX

The following functions \(f(x)\) [\(x\) is a real variable] can be represented by series
(Taylor or Laurent) in powers of \((x-2)\). Find the radius of convergence of the series in
each case
\[
\begin{array}{rcl}
\text{(a)}\quad f(x)&=&e^{x^2},\\[0.6ex]
\text{(b)}\quad f(x)&=&\dfrac{\sin x-3}{\sin x-2},\\[1.2ex]
\text{(c)}\quad f(x)&=&\dfrac{e^x}{(x-2)(1+e^x)}.
\end{array}
\]

Topic Tags

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