Q.Find the coefficient of $x^6$ in the expansion of $e^{2x}$ using series expansion.

Certain important functions can be expressed as an infinite sum of powers of $x$, called a power series; well-known examples include $e^x=1+\dfrac{x}{1!}+\dfrac{x^2}{2!}+\cdots$, $e^{-x}=1-\dfrac{x}{1!}+\dfrac{x^2}{2!}-\cdots$, $\sin x=x-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\cdots$, $\cos x=1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}-\cdots$, and, for $|x|<1$, $\log(1+x)=x-\dfrac{x^2}2+\dfrac{x^3}3-\cdots$. These expansions are proved rigorously using more advanced tools met later in a full calculus course, so at this stage they are simply stated and used as given facts. A common application is extracti …