Exercise 6.4.9

Question

Let
  $$
  h(x)=\sum_{n=1}^{\infty} \frac{1}{x^{2}+n^{2}}
  $$
  \begin{enumerate}[label=(\alph*)] 
  \item Show that $h$ is a continuous function defined on all of $\mathbf{R}$.
  \item Is $h$ differentiable? If so, is the derivative function $h^{\prime}$ continuous?
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
    \item Use the M-Test with $M_n = \frac{1}{n^2}$.
    \item The termwise derivatives are
    $$h'_n(x) = \frac{-2x}{x^4 + 2x^2 n^2 + n^4}$$
    with $\left|h'_n(x)\right| < 2x / n^4$. For any fixed $a > 0$, over the interval $(-a, a)$ we can bound $\left|h'_n(x)\right|$ with $M_n = 2a / n^4$, so by the Differentiable Limit Theorem as well as uniform convergence via the M-Test we have that $h$ is differentiable with $h'$ continuous.
   \end{enumerate}

Exercise 6.4.9

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Exercise 6.4.9

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