Strategy for finding Maclaurin series I know the series for $\cos(x)$ it is $\sum \limits_{n=0}^\infty \dfrac{(-1)^n x^{2n}}{(2n)!}$ 
which will result in $\sum \limits_{n=0}^\infty \dfrac{\left(-1\right)^ n x^{2n+1}}{\left(2n\right)!}$ 
Which is great when you already know the series; however, my question is how does one find Maclaurin series when you don't already know the series? 
 A: If you know that your'e function satisfies a differential equation you can use Picard iteration to find the McLaurin series. If your functions satisfy $y' = f(t,y)$, you can use the formula $ y(x) = y(0) + \int_0^x f(t,y(t)) dt$ to calculate the series.
A: Are you trying to find the Maclaurin series for $x\cos(x)$?  If so, what you've done is valid.
Practically, it is much easier to find the Maclaurin series representing a given function by relating it to some function with known Maclaurin series, as you've done here.  In general, you must take care to note where the series converges, but here your radius of convergence will be infinite.
You can also use the following formula to calculate the Maclaurin series:
$$
\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n
$$
This is valid whenever $f(x)$ is infinitely differentiable in a neighborhood of $0$.  Whether or not $f(x)$ is equal to its Maclaurin series is another question, but in your case, it is.  Using this formula to find Maclaurin series is often very difficult, as finding patterns for the derivatives of $f(x)$ is not always clear.  In the case of $\cos(x)$, the pattern is fairly clear, so for your question, it is a natural starting point.
