Taylor's theorem is a mathematical result that allows us to approximate a function using a polynomial expansion. It states that any sufficiently smooth function can be approximated by a polynomial that is centered around a specific point. This theorem is particularly useful in calculus and numerical analysis.

Taylor's Theorem:
Let f(x) be a function that is infinitely differentiable in an interval I surrounding a point a. Then, for any positive integer n, the function f(x) can be approximated by the nth degree Taylor polynomial:

f(x) ≈ Pn(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)²/2! + ... + fⁿ⁽ⁿ⁾(a)(x-a)ⁿ/ⁿ!

Proof:
To prove Taylor's theorem, we start by considering the function f(x) and its derivatives evaluated at the point a. Let's denote the derivatives as f'(a), f''(a), f'''(a), and so on.

Step 1: Zeroth Degree Taylor Polynomial
The zeroth degree Taylor polynomial is simply the function value at a:

P₀(x) = f(a)

Step 2: First Degree Taylor Polynomial
The first degree Taylor polynomial includes the linear term of the function:

P₁(x) = f(a) + f'(a)(x-a)

Step 3: Second Degree Taylor Polynomial
The second degree Taylor polynomial includes the quadratic term of the function:

P₂(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)²/2!

Step k: kth Degree Taylor Polynomial
Following the same pattern, we can derive the kth degree Taylor polynomial:

Pₖ(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)²/2! + ... + fᵏ(a)(x-a)ᵏ/ᵏ!

Step n: nth Degree Taylor Polynomial
Finally, when we reach the nth degree Taylor polynomial, we have:

Pₙ(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)²/2! + ... + fⁿ⁽ⁿ⁾(a)(x-a)ⁿ/ⁿ!

Approximation:
The nth degree Taylor polynomial becomes an increasingly accurate approximation of the original function f(x) as n approaches infinity. This is because the additional terms in the polynomial capture the behavior of the higher-order derivatives of f(x) at point a.

By using Taylor's theorem, we can approximate the value of a function at a specific point without needing to evaluate the function directly. This is especially useful when dealing with complex functions or when numerical methods are required.

Note: Taylor's theorem also provides a remainder term, known as the Taylor remainder, which quantifies the error between the original function f(x) and its nth degree Taylor polynomial approximation. However, the proof of the remainder term involves more advanced mathematical concepts such as the concept of a limit.