A Taylor series is a mathematical representation of a function as an infinite sum of terms that are calculated from the values of that function's derivatives at a single point. It is used to approximate the behavior of a function near a specific point.
Taylor series can be used to evaluate limits by substituting the limit value into the series and then taking the limit as the number of terms approaches infinity. This allows for the approximation of the value of a function at a specific point, even if the function itself is not defined at that point.
A Taylor series is a representation of a function at a specific point, while a Maclaurin series is a special case of a Taylor series where the point is at x=0. In other words, a Maclaurin series is a Taylor series centered at 0.
Taylor series can be used in many areas of mathematics and science, including calculus, physics, and engineering. They are particularly useful in approximating complex functions and solving differential equations.
No, Taylor series can only be used to evaluate limits for functions that are continuous and differentiable at the point of evaluation. If a function is not continuous or differentiable at a specific point, then Taylor series cannot be used to evaluate its limit at that point.