The Taylor-series remainder for $f\left(x\right)\=e^x$ is

For each fixed $x$, ${\stackrel{\^}{R}}_{n\+1}\left(x\right)\to 0$, as suggested by Figure 8.5.2(a) where $n$ is controlled by the slider.

As $n\to \infty$, ${\stackrel{\^}{R}}_{n\+1}\left(x\right)\to 0$ but the limit is not uniform in $x$. As $n$ increases, the interval on which the remainder gets close to zero increases to the right.

In the limit as $n\to \infty$, the interval on which the remainder approaches zero becomes the whole real line.

The Maclaurin series is $f\left(x\right)\=e^x\=\sum _{n\=0}^{\infty }\frac{x^n}{n\!}$ because $f^{\left(n\right)}\left(x\right)\=e^x$ and $f^{\left(n\right)}\left(0\right)\=1$ for $n\=0\,1\,\dots \.$

Write $R\=\dots$
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Write the exponential function, being sure to use a Maple template or palette for "e".

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Complete the dialog as per Figure 8.5.2(b).$$

Write $f\left(x\right)\=\dots$(Use the correct "e"!)
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