The Taylor power-series for a function $f$ about $x=0$ converges to $f$ for $-1\leq x\leq 1$. The $n$th-degree Taylor polynomial for $f$ about $x=0$ is given by $P_n(x)=\displaystyle\sum_{k=1}^n (-1)^k \dfrac{x^k}{k^2+k+1}$. Of the following, which is the smallest number $M$ for which the alternating power-series error bound guarantees that $|f(1)-P_4(1)|\leq M$ ?

$|f(1)-P_4(x)|$ describes the difference between the actual value versus the value approximated by the Taylor power-series. This error margin will always be less than then $a_{N+1}$, where $N$ is the number of terms used in the approximation. In the question, we are using $P_4(1)$, which is the first four terms of the power-series. Therefore, the absolute value of the remainder will be less than the value of the 5^th term.