8. Recall that $\cos x=\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2 n}}{(2 n) !}$.
(a) Write a power series for $g(x)=x^{2} \cos \left(x^{2}\right)$.
(b) Write a power series for $g^{\prime}(x)$. (You should not need the product rule or chain rule.)
(c) Write a power series for $G(x)=\int x^{2} \cos \left(x^{2}\right) d x$, given $F(0)=3$. (You should not need any special methods of integration.)
Answer
The power series for \(G(x)=\int x^{2} \cos \left(x^{2}\right) d x\) with \(F(0)=3\) is \[G(x) = -\frac{x^{39}}{249692574523392000} + \frac{x^{35}}{732297646080000} - \frac{x^{31}}{2702527027200} + \frac{x^{27}}{12933043200} - \frac{x^{23}}{83462400} + \frac{x^{19}}{766080} - \frac{x^{15}}{10800} + \frac{x^{11}}{264} - \frac{x^{7}}{14} + \frac{x^{3}}{3} + 3\]
Step 1 ：Given the power series for \(\cos x\) as \(\cos x=\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2 n}}{(2 n) !}\), we can substitute \(x^2\) into the power series for \(\cos x\) and then multiply the result by \(x^2\) to get the power series for \(g(x)=x^{2} \cos \left(x^{2}\right)\).
Step 2 ：The power series for \(g(x)=x^{2} \cos \left(x^{2}\right)\) is \[g(x) = x^2\left(-\frac{x^{36}}{6402373705728000} + \frac{x^{32}}{20922789888000} - \frac{x^{28}}{87178291200} + \frac{x^{24}}{479001600} - \frac{x^{20}}{3628800} + \frac{x^{16}}{40320} - \frac{x^{12}}{720} + \frac{x^{8}}{24} - \frac{x^{4}}{2} + 1\right)\]
Step 3 ：We can differentiate the power series for \(g(x)\) term by term to get the power series for \(g^\prime(x)\).
Step 4 ：The power series for \(g^\prime(x)\) is \[g^\prime(x) = x^2\left(-\frac{x^{35}}{177843714048000} + \frac{x^{31}}{653837184000} - \frac{x^{27}}{3113510400} + \frac{x^{23}}{19958400} - \frac{x^{19}}{181440} + \frac{x^{15}}{2520} - \frac{x^{11}}{60} + \frac{x^{7}}{3} - 2x^{3}\right) + 2x\left(-\frac{x^{36}}{6402373705728000} + \frac{x^{32}}{20922789888000} - \frac{x^{28}}{87178291200} + \frac{x^{24}}{479001600} - \frac{x^{20}}{3628800} + \frac{x^{16}}{40320} - \frac{x^{12}}{720} + \frac{x^{8}}{24} - \frac{x^{4}}{2} + 1\right)\]
Step 5 ：We can integrate the power series for \(g(x)\) term by term and then add the constant \(F(0)=3\) to get the power series for \(G(x)=\int x^{2} \cos \left(x^{2}\right) d x\).
Step 6 ：The power series for \(G(x)=\int x^{2} \cos \left(x^{2}\right) d x\) with \(F(0)=3\) is \[G(x) = -\frac{x^{39}}{249692574523392000} + \frac{x^{35}}{732297646080000} - \frac{x^{31}}{2702527027200} + \frac{x^{27}}{12933043200} - \frac{x^{23}}{83462400} + \frac{x^{19}}{766080} - \frac{x^{15}}{10800} + \frac{x^{11}}{264} - \frac{x^{7}}{14} + \frac{x^{3}}{3} + 3\]
