$$x - rac{x^2}{2!} + rac{x^3}{3!} - rac{x^4}{4!} + ... - rac{x^n}{n!}$$

[Taylor Series Expansion]

The given series is the Taylor series expansion of the exponential function $e^x$ around $x = 0$. $$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + ... + \frac{x^n}{n!} $$

[Proof of the Taylor Series]

The Taylor series expansion of a function $f(x)$ around a point $a$ is given by:

$$f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ...$$

where $f'(a)$, $f''(a)$, $f'''(a)$, ... are the first, second, third, ... derivatives of $f(x)$ evaluated at $x = a$.

[Proof for the Exponential Function]

Let's prove the Taylor series expansion for the exponential function $e^x$.

1. First Derivative:

$$f(x) = e^x, \quad f'(x) = \frac{d}{dx} e^x = e^x$$

1. Second Derivative:

$$f''(x) = \frac{d}{dx} e^x = e^x$$

1. Third Derivative:

$$f'''(x) = \frac{d}{dx} e^x = e^x$$

and so on.

It can be seen that the $n$th derivative of $e^x$ is $e^x$ for all $n$.

1. Evaluating at $x = 0$:

$$f(0) = e^0 = 1, \quad f'(0) = e^0 = 1, \quad f''(0) = e^0 = 1, \quad \ldots$$

1. Substituting into the Taylor Series Formula:

$$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + ... + \frac{x^n}{n!}$$

[Convergence of the Taylor Series]

The Taylor series for the exponential function converges for all values of $x$. This can be shown using the ratio test:

$$\lim_{n\to\infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n\to\infty} \left| \frac{\frac{x^{n+1}}{(n+1)!}}{\frac{x^n}{n!}} \right| = \lim_{n\to\infty} \left| \frac{x}{n+1} \right| = 0$$

Since the limit is less than 1 for all $x$, the series converges for all $x$.

[Applications of the Taylor Series Expansion]

The Taylor series expansion of the exponential function has many applications in mathematics, physics, and engineering. Some examples include:

1. Approximating the value of $e^x$ for small values of $x$.

2. Solving differential equations.

3. Finding the eigenvalues and eigenvectors of matrices.

4. Computing integrals and derivatives.