EnglishMultivariable CalculusTaylor Series
We can define a polynomial which approximates a smooth function in the vicinity of a point with the following idea: match as many derivatives as possible.
The utility of this simple idea emerges from the convenient simplicity of polynomials and the fact that a wide class of functions look pretty much like polynomials when you zoom in around a given point.
First, a bit of review on the exponential function : we define to be the function which maps 0 to 1 and which is everywhere equal to its own derivative. It follows (nontrivially) from this definition that , so may define and write the exponential function as . The value of is approximately 2.718.
Example
Find the quadratic polynomial whose zeroth, first, and second derivatives at the origin match those of the exponential function.
Solution. Since is quadratic, we must have
for some and . To match the
For , we calculate , so to get , we have to let . So
is the best we can do. Looking at the figure, we set that does indeed do a better job of 'hugging' the graph of near than the best linear approximation () does.
The best constant, linear, and quadratic approximations of XEQUATIONX4191XEQUATIONX near the origin
We can extend this idea to higher order polynomials, and we can even include terms for all powers of , thereby obtaining an infinite series:
Definition (Taylor Series)
The Taylor series, centered at , of an infinitely differentiable function is defined to be
Example
Find the Taylor series centered at the origin for the exponential function.
Solution. We continue the pattern we discovered for the quadratic approximation of the exponential function at the origin: the $n$th derivative of is , while the $n$th derivative of the exponential function is at the origin. Therefore, , and we obtain the Taylor series
It turns out that this series does in fact converge to , for all .
Taylor series properties
It turns out that if the Taylor series for a function converges, then it does so in an interval centered around . Furthermore, inside the interval of convergence, it is valid to perform term-by-term operations with the Taylor series as though it were a polynomial:
- We can multiply or add Taylor series term-by-term.
- We can integrate or differentiate a Taylor series term-by-term.
- We can substitute one Taylor series into another to obtain a Taylor series for the composition.
Theorem
All the operations described above may be applied wherever all the series in question are convergent. In other words, and have Taylor series and converging to and in some open interval, then the Taylor series for , , , and converge in that interval and are given by , , , and , respectively. If has an infinite radius of convergence, then the Taylor series for is given by .
The following example shows how convenient this theorem can be for finding Taylor series.
Example
Find the Taylor series for centered at .
Solution. Taking many derivatives is going to be no fun, especially with that second term. What we can do, however, is just substitute into the Taylor series for the exponential function, multiply that by , and add the Taylor series for cosine:
In summation notation, we could write this series as where is equal to if is even and if is odd.
Exercise
Find the Taylor series for centered at the origin, and show that it converges to for all .
Use your result to find . Hint: think about differentiation.
Solution. Calculating derivatives of , we find that the Taylor series centered at the origin is . Furthermore, we know that
for , by the formula for infinite geometric series.
We can use this result to find by differentiating both sides and multiplying both sides by :
We get
Exercise
Show that is equal to by showing that .
Solution. Integrating the equation
term by term, we find that
Substituting gives
Each of the terms other than the
