The absolute value of the Taylor-series remainder for $f\left(x\right)\=\ln \left(1\+x\right)$ is

For each fixed $x$, $\left|{\stackrel{\^}{R}}_{n\+1}\left(x\right)\right|\to 0$, as suggested by Figure 8.5.7(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.

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

Calculus palette: Limit template
Add the positivity assumption!
Context Panel: Evaluate and Display Inline

Write the given function.

Context Panel: Series≻Formal Power Series
Complete the dialog as per Figure 8.5.7(b).$$

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

Expression palette: Summation template
Calculus palette: nth-derivative template

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

Context Panel: Evaluate and Display Inline

Context Panel: Simplify≻Assuming Integer