Taylor Series — Question 10

Question 10 – Solution

Step 1: Encode the product as a real part.

$$e^x\cos x=\operatorname{Re}\left(e^x e^{ix}\right)=\operatorname{Re}\left(e^{(1+i)x}\right).$$

Step 2: Expand the complex exponential.

$$e^{(1+i)x}=\sum_{n=0}^{\infty}\frac{(1+i)^n x^n}{n!},$$ so the real coefficient of $x^n$ is $\operatorname{Re}((1+i)^n)/n!$. Since $1+i=\sqrt2e^{i\pi/4}$, $$\boxed{e^x\cos x=\sum_{n=0}^{\infty}\frac{2^{n/2}\cos(n\pi/4)}{n!}x^n}.$$

Step 3: Extract nonzero terms.

The $x^2$ coefficient vanishes. The first four nonzero terms are $$\boxed{1+x-\frac{x^3}{3}-\frac{x^4}{6}}\qquad\left(\text{next term }-\frac{x^5}{30}\right).$$ The factorial denominator gives infinite radius, so the series represents the function for every real $x$.