Taylor Series for Exponential Function Centered at a2

Find the Taylor series for the function $f(x) = e^x$ centered at $a = 2$.

The Taylor series expansion is a powerful tool in calculus to express functions as infinite sums of terms based on their derivatives at a particular point. In this problem, you are tasked with finding the Taylor series for the exponential function $e^x$, centered not at zero, the typical point for a Maclaurin series, but at $a = 2$. This problem tests understanding of how Taylor series can represent functions around different points, not just the origin, demonstrating their flexibility and utility in approximating functions in specific intervals.

To solve this problem, consider the standard formula for a Taylor series centered at a point $a$, which involves the $n$-th derivatives of the function evaluated at that point, divided by $n!$, and multiplied by $(x - a)^n$. The exponential function $e^x$ is particularly interesting because its derivatives are all the same, $e^x$, which makes the computation straightforward, yet this problem introduces subtle complexity by shifting the center to a non-zero point. Such an exploration provides insight into how to adapt known strategies to solve variations of problems, a vital skill in calculus and broader mathematical problem solving.

Understanding how to generate Taylor series for functions like $e^x$ around different points enriches your toolkit for approximation, enabling you to apply these concepts confidently to physical problems, especially where evaluating the function at non-standard points leads to more convenient or accurate models.