The Maclaurin series for \(\cos(x)\) is given by:

\(\cos(x)=\sum_{n=0}^\infty(-1)^n\frac{x^{2n}}{(2n)!}=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}-\frac{x^{10}}{10!}+...\)

Multiply the Maclaurin series for \(\cos(x)\text{ by }x^4\) and obtain the Maclaurin series for \(f(x)=x^4\cos(x)\) as shown below.

\(x^4\cos(x)=x^4(\sum_{n=0}^\infty(-1)^n\frac{x^{2n}}{(2n)!})\)

\(=\sum_{n=0}^\infty(-1)^n\frac{x^4\cdot x^{2n}}{(2n)!}\)

\(=\sum_{n=0}^\infty(-1)^n\frac{x^{2n+4}}{(2n)!}\)

Thus, the Maclaurin series for \(f(x)=x^4\cos(x)\) is,

\(f(x)=\sum_{n=0}^\infty(-1)^n\frac{x^{2n+4}}{(2n)!}\)

Now find the third term for this series as shown below.

\(f(x)=\sum_{n=0}^\infty(-1)^n\frac{x^{2n+4}}{(2n)!}\)

\(=x^4-\frac{x^6}{2!}+\frac{x^8}{4!}-\frac{x^{10}}{6!}+\frac{x^{12}}{8!}-...\)

\(=x^4-\frac{x^6}{2}+\frac{x^8}{24}-\frac{x^{10}}{720}+\frac{x^{12}}{40320}-...\)

Hence, the third term of the series is \(\frac{x^8}{24}ZS

\(\cos(x)=\sum_{n=0}^\infty(-1)^n\frac{x^{2n}}{(2n)!}=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}-\frac{x^{10}}{10!}+...\)

Multiply the Maclaurin series for \(\cos(x)\text{ by }x^4\) and obtain the Maclaurin series for \(f(x)=x^4\cos(x)\) as shown below.

\(x^4\cos(x)=x^4(\sum_{n=0}^\infty(-1)^n\frac{x^{2n}}{(2n)!})\)

\(=\sum_{n=0}^\infty(-1)^n\frac{x^4\cdot x^{2n}}{(2n)!}\)

\(=\sum_{n=0}^\infty(-1)^n\frac{x^{2n+4}}{(2n)!}\)

Thus, the Maclaurin series for \(f(x)=x^4\cos(x)\) is,

\(f(x)=\sum_{n=0}^\infty(-1)^n\frac{x^{2n+4}}{(2n)!}\)

Now find the third term for this series as shown below.

\(f(x)=\sum_{n=0}^\infty(-1)^n\frac{x^{2n+4}}{(2n)!}\)

\(=x^4-\frac{x^6}{2!}+\frac{x^8}{4!}-\frac{x^{10}}{6!}+\frac{x^{12}}{8!}-...\)

\(=x^4-\frac{x^6}{2}+\frac{x^8}{24}-\frac{x^{10}}{720}+\frac{x^{12}}{40320}-...\)

Hence, the third term of the series is \(\frac{x^8}{24}ZS