Answer
$$3.390842014 \times 10^{-7}$$
Work Step by Step
Since
\begin{array}{ll}
{f(x)=\cos x,} & {f(0)=1} \\
{f^{\prime}(x)=-\sin x,} & {f^{\prime}(0)=0} \\
{f^{\prime \prime}(x)=-\cos x,} & {f^{\prime \prime}(0)=-1} \\
{f^{\prime \prime \prime}(x)=\sin x,} & {f^{\prime \prime \prime}(0)=0} \\
{f^{(4)}(x)=\cos x,} & {f^{(4)}(0)=1} \\
{f^{(5)}(x)=-\sin x,} & {f^{(5)}(0)=0}
\end{array}
Then $$T_{5}(x)=1-\frac{1}{2} x^{2}+\frac{1}{24} x^{4}$$
Since
$$ \left|f(x)-T_{n}(x)\right| \leq K \frac{|x-a|^{n+1}}{(n+1) !} $$
Then
\begin{aligned}
\left|f(0.25)-T_{5}(0.25)\right| &\leq(1) \frac{|0.25-0|^{5+1}}{(5+1) !} \\
\left|f(3.9)-T_{2}(3.9)\right| & \leq \frac{(0.25)^{6}}{6 !} \\
&=\frac{(0.25)^{6}}{720} \\
& \approx \frac{(3.25)^{6}}{720}
\end{aligned}
To check
\begin{aligned}
\left|f(0.25)-T_{5}(0.25)\right| & \approx|0.9689124217-0.9689127604| \\
& \approx 3.387 \times 10^{-7}<3.390842014 \times 10^{-7}
\end{aligned}
