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

For each fixed $x$, ${\stackrel{\^}{R}}_{n\+1}\left(x\right)\to 0$, as suggested by Figure 8.5.5(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 and left.

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

Write $R\=\dots$
Context Panel: Assign Name

Calculus palette: Limit template
Context Panel: Evaluate and Display Inline

Write $\sinh \left(x\right)$

Context Panel: Series≻Formal Power Series
Complete the dialog as per Figure 8.5.5(b).$$

Write $f\left(x\right)\=\dots$
Context Panel: Assign Function

Expression palette: Summation template

Context Panel:
2-D Math≻Convert To≻Inert Formb

Context Panel: Evaluate and Display Inline

Control-drag the summand and press the Enter key.

Context Panel: Evaluate at a Point≻$n\=2\cdot k\+1$ 

Context Panel: Expand≻Expand

Context Panel: Simplify≻Assuming Integer