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Power series expansion is a method used in mathematics to express functions as an infinite sum of terms calculated from the function's derivatives at a single point. It's like breaking down a complex function into simpler building blocks that we can easily work with. A power series for a function can be represented as:

• \( f(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + \ldots \)

Each term consists of a coefficient and a power of \(x\). This approach is particularly useful when finding limits or derivatives. It helps in approximating functions that are otherwise difficult to handle. In the context of our exercise, the expansion helps simplify the expression \(\frac{e^x - 1 - x}{x^2}\) into more manageable terms. Expanding functions into power series can turn difficult calculus problems into straightforward arithmetic.

A Taylor series is a specific kind of power series expansion of a function about a point. It approximates functions with a polynomial that incorporates information from the function’s derivatives at a single point. The formula for the Taylor series of a function \(f(x)\) centered at \(a\) is:

• \(f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \ldots \)

For most functions, if enough terms of the series are used, the Taylor series converges to the actual function. In our exercise, the exponential function \(e^x\) is approximated using its Taylor series, which simplifies calculations. This is because the series provides an approximation of \(e^x\) around \(x = 0\), allowing us to easily find and evaluate limits and derivatives.

Exponential function expansion involves expressing the function \(e^x\) as an infinite series of terms, making it much easier to use in a variety of calculus problems. The exponential function \(e^x\) has a well-known expansion derived from the Taylor series, which looks like:

• \(e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \ldots\)

This expansion is particularly useful as it converges to the actual value of \(e^x\) for all \(x\). In the given exercise, expanding \(e^x\) is the first step. This allows us to subtract terms like \(1 + x\) efficiently, leading to a simplified expression that can be analyzed further using other calculus techniques like limits.

Calculus limits are a fundamental concept that describe the behavior of a function as it approaches a particular point. Limits help us understand the tendencies of a function's value as the input approaches some number. In our current exercise, the limit we are evaluating is:

• \(\lim_{x \to 0} \frac{e^x - 1 - x}{x^2}\)

Calculating this involves finding the value that the function approaches as \(x\) gets closer to 0. By using the power series for the function, we simplify the limit problem to just evaluating the constant term as other terms disappear when \(x\) is zero. Thus, the limit of the expression becomes \(\frac{1}{2}\), based solely on the constant from the expanded series.