Define the function
\begin{equation*}
g(x) = \begin{cases}
\mathrm{e}^{x^{-2}} &\text{if } x \gt 0
\\
0 &\text{if } x \leq 0
\end{cases}\,,
\end{equation*}
and notice that $g$ is infinitely differentiable,
but has no Taylor series centered at $x=0$
since $g^{(n)}(0) = 0$ for all $n \gt 0$.
Furthermore $g$ is monotonically increasing,
and $\lim_{x \to -\infty} g = 0$
and $\lim_{x \to \infty} g = 1$.
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-
Using the function $g$, define an infinitely differentiable function $p(x)$
such that $p(x) \gt 0$ for $x \in (0,1)$ but $p(x) = 0$ otherwise.
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Using the function $g$, define an infinitely differentiable function $q(x)$
such that $q(x) = 0$ for $x \leq 0$,
is strictly increasing for $x \in [0,1]$,
and $q(x) = 1$ for $x \geq 1$.
(This one is called a smooth step function
and is very important in differential geometry and in computer programming.)
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Verify Leibniz’s famous identity
\begin{equation*}
\sqrt{6} = \sqrt{1+\sqrt{-3}} + \sqrt{1-\sqrt{-3}}
\,.
\end{equation*}
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Using the Taylor series for $\arctan(x)$ one can prove that
\begin{equation*}
\frac{\pi}{4} = 1 - \frac13 + \frac15 - \frac17 + \dotsb
\,.
\end{equation*}
Use this to prove that
\begin{equation*}
\frac{\pi}{8} = \frac{1}{1\cdot 3} +\frac{1}{5\cdot 7} +\frac{1}{9\cdot 11} + \dotsb
\,.
\end{equation*}