GATE 2024 EE Q.31: solution and answer

Official answer: 0.0Source: Official answer key, GATE 2024 EE (IISc Bengaluru)

1. Maclaurin series of cosine: $\cos z=1-\frac{z^2}{2!}+\frac{z^4}{4!}-\frac{z^6}{6!}+\cdots ={\sum }_{n\ge 0}\frac{(-1{)}^n{z}^{2n}}{(2n)!}$. Only even powers of $z$ appear.

2. Maclaurin series of the exponential with $w=z^2$: $e^{z^2}={\sum }_{n\ge 0}\frac{(z^2{)}^n}{n!}=1+z^2+\frac{z^4}{2!}+\frac{z^6}{3!}+\cdots$. Again only even powers appear.

3. Both series converge for every $z$, so we can add them term by term: $$f(z)=2+\left(1-\frac{1}{2}\right)z^2+\left(\frac{1}{24}+\frac{1}{2}\right)z^4+\cdots =2+\frac{1}{2}{z}^2+\frac{13}{24}{z}^4+\cdots$$

4. Parity check: $\cos (-z)=\cos z$ and $e^{(-z{)}^2}=e^{z^2}$, so $f(-z)=f(z)$. $f$ is an even function, so every odd-power coefficient in its Taylor series about $0$ is zero.

5. So the coefficient of $z^5$ is $0$, which is $0.0$ to 1 decimal place.