Find the Taylor series expansion for ex when x is zero, and determine its radius of convergence.
Before starting this problem, note that the Taylor series expansion of any function about the point c = 0 is the same as finding its Maclaurin series expansion.
Step 1: Find Coefficients
Let f(x) = ex. To find the Maclaurin series coefficients, we must evaluate
for k = 0, 1, 2, 3, 4, ....
Because f(x) = ex, then all derivatives of f(x) at x = 0 are equal to 1. Therefore, all coefficients of the series are equal to 1.
Step 2: Substitute Coefficients into Expansion
By substitution, the Maclaurin series for ex is
Step 3: Radius of Convergence
The ratio test gives us:
Because this limit is zero for all real values of x, the radius of convergence of the expansion is the set of all real numbers.
Explanation of Each Step
Maclaurin series coefficients, ak are always calculated using the formula
where f is the given function, and in this case is e(x). In step 1, we are only using this formula to calculate coefficients. We found that all of them have the same value, and that value is one.
Step 2 was a simple substitution of our coefficients into the expression of the Taylor series given on the previous page.
This step was nothing more than substitution of our formula into the formula for the ratio test. Because we found that the series converges for all x, we did not need to test the endpoints of our interval. If however we did find that the series only converged on an interval with a finite width, then we may need to take extra steps to determine the convergence at the boundary points of the interval.
What are we Doing?
The following video provides a graphical interpretation of the Taylor approximation to ex about the point c = 3. In the derivation above, we considered an expansion at c = 0. The instructor uses the term "Taylor approximation" in the same way we use the term "Taylor expansion".
|Visualizing Taylor Series for e^x|
|A graphical description of the Taylor series of ex.|
|This video can be found on the Kahn Academy website, and carries a Creative Commons copyright (CC BY-NC-SA 3.0).|
In this example, we found the Macluarin expansion of the exponential function.
|The Maclaurin Expansion of ex|
| The Maclaurin series expansion for ex is given by
This formula is valid for all real values of x.