Write the first four terms of the power series in x

Write the first four terms of the power series in $x$ for $e^{\cos x}$.

Explanation of Method

Writing $\cos x=1+u$ with $u=\cos x-1=-\frac{x^2}2+\frac{x^4}{24}-\cdots$, then $e^{\cos x}=e\cdot e^u$; expanding $e^u=1+u+u^2/2+u^3/6+\cdots$ and collecting like powers of $x$ builds the series order by order.

Set $u=\cos x-1$ and expand $e^u=1+u+\frac{u^2}2+\frac{u^3}6+\cdots$.
Collect powers of $u$ in terms of $x$ up through $x^6$.
Multiply the result by $e$.

Step 1 · Expand $u$ and its powers

$$ u=-\frac{x^2}2+\frac{x^4}{24}-\frac{x^6}{720}+\cdots, \qquad u^2=\frac{x^4}4-\frac{x^6}{24}+\cdots, \qquad u^3=-\frac{x^6}8+\cdots $$

Step 2 · Collect coefficients of $e^u$

$$ e^u = 1+u+\frac{u^2}2+\frac{u^3}6+\cdots $$ Coefficient of $x^2$: $-\frac12$. Coefficient of $x^4$: $\frac1{24}+\frac12\cdot\frac14=\frac16$. Coefficient of $x^6$: $-\frac1{720}-\frac1{48}-\frac1{48}=-\frac{31}{720}$.

Step 3 · Multiply by $e$

$$ e^{\cos x} = e\left(1-\frac{x^2}2+\frac{x^4}6-\frac{31x^6}{720}+\cdots\right) $$

Summary Table

Property	Value
First four terms	$e,\ -\frac{ex^2}2,\ \frac{ex^4}6,\ -\frac{31ex^6}{720}$
At $x=0$	$e^{\cos0}=e$
Region of validity	$\mathbb R$

Additional Notes

Only even powers of $x$ appear, since $\cos(-x)=\cos x$ makes $e^{\cos x}$ an even function.
Every coefficient carries a factor of $e$, since $e^{\cos x}=e\cdot e^{\cos x-1}$ and the second factor is a genuine Maclaurin series in $x$ starting at $1$.
Because $\cos x$ ranges over $[-1,1]$, the function oscillates between $e^{-1}$ (at $x=(2k+1)\pi$) and $e$ (at $x=2k\pi$), and the series here captures the behavior near its maximum at $x=0$.

Result

$$ e^{\cos x} = e-\frac{ex^2}2+\frac{ex^4}6-\frac{31ex^6}{720}+\cdots, \quad x\in\mathbb R. $$