Maclaurin Series for Exponential Function

Find the Maclaurin series for the function $f(x) = e^x$.

The Maclaurin series is a way to represent functions as infinite sums of their derivatives at a single point, usually zero. This problem asks us to find the Maclaurin series for the exponential function, e to the power of x. A key concept here is understanding how to expand a function into an infinite series by evaluating derivatives at zero, and recognizing where the exponential function stands out because its derivatives are constant and identical to the function itself.

To derive the Maclaurin series for $e^x$, one should start by recalling that the formula for a Maclaurin series is derived from Taylor's series centered at zero. It involves evaluating the function and its successive derivatives at zero and using these in a power series representation. The exponential function is unique because each derivative of $e^x$ is itself, which simplifies our work considerably. Thus, for each derivative at zero, the result is always one, making the series a straightforward summation of $x^n$ divided by $n!$, where $n$ starts from zero and goes to infinity.

Understanding how functions can be represented as series is foundational in calculus, and this particularly elegant series for the exponential function highlights the broader utility and beauty of power series. Such representations are not just mathematical abstractions; they are essential in approximating functions during computations in both pure and applied contexts, such as physics or engineering, where they aid in simplifying complex mathematical models.