I would like to know whether the following power series converges or diverges.
$$1-x+\frac{x^2}{2!}-\frac{x^3}{3!} + \frac{x^4}{4!} + \cdots.$$
My intutition tells me that for any nonzero $x$, the series diverges, but I am not sure how to verify it
How should I verify it?
The series does converge for all $x\in\mathbb{R}$, and it is the series expression for $e^{-x}$. First of all, notice that $\lim_{n\to\infty}(n!)^{1/n}=\infty$ (a proof of this fact can be found here). So the radius of convergence of the above power series is \begin{eqnarray} R=\frac{1}{\lim_{n\to\infty}(1/n!)^{1/n}}=\lim_{n\to\infty} (n!)^{1/n}=\infty. \end{eqnarray} Therefore the series converges for all $x\in\mathbb{R}$. Now compare the series with the talyor series expression for $e^{-x}$.
If $n+1\gt x$ you have $\frac {x^n}{n!}\gt \frac{x^{n+1}}{(n+1)!}$
Ignoring the finite number of initial terms, and with a little more work, you can use the alternating series test to show that this converges (you have to show the magnitude of the terms goes to zero).
