0. Can we approximate a function using a power series?
With the calculus’ help, we can!
1. Consider the function
We suppose it is smooth (i.e., has infinitely many derivatives). We see
We also have
The general case appears to be
Thus the coefficients are
and we can reconstruct the function .
2. Definition. Let be any function. The “Taylor Series about ” is a series
When , it’s called a MacLaurin Series.
Remark 1. If we have the MacLaurin series for a function, and if the series converges for any value of , then we can use the MacLaurin series as a synonym for the original function. That’s the usefulness of MacLaurin series.
Remark 2. The Taylor series helps us compute when and when is known. For example, can be computed using the Taylor series of about .
Example 1. Consider the function . We see that
Thus the MacLaurin series for the exponential function is
Where does this series converge?
Using the absolute ratio test, we find
So the series converges for any value of .
Example 2. Consider the sine function . What is its MacLaurin series? Writing
The coefficients are sort of “periodic” in the sense that only the odd ones remain, and their sign alternates. We have
thus the MacLaurin series is
Where does this converge?
Using the ratio test, we have
Thus it converges for any value of .
Example 3. Lets find the MacLaurin series for . We see that
we need to find the . We see that
So we have the MacLaurin series be
Where will it converge? Using the absolute ratio test, we have
Thus the MacLaurin series for converges for any value of .
3. Euler’s Formula. Recall Euler’s formula states
and we have the MacLaurin series for . Setting , we find
Gathering the real and imaginary parts together we get
But look: the imaginary part is precisely the MacLaurin series for ! And the real part is the MacLaurin series for , too! So what did we do? We just derived Euler’s formula.
4. Taylor Series Makes Approximations Consider the function
Its Taylor series about is the same as the MacLaurin series for the function
Just write . For small , we can use the first couple of terms in the Taylor series as an approximate value. So what’s the first 6 nonzero terms in the Taylor series? We see
describes the constant term. The linear term has coefficient
The quadratic term
Observe how the exponent behaves when differentiating: . The numerator is always odd, the denominator doesn’t change. So the derivative would be
This gives us our coefficients! We then have
for . Observe the numerator appears odd, but we can justify it thus:
At any rate, this gives us a polynomial expression for as
where is the “error” term. What does the error term tell us? Precisely how good or bad our polynomial approximates . That is to say, how approximates .
Question: how do we find the error term?
Exercise 1 [Math.SE]. Using the MacLaurin series for sine and cosine, check that
Exercise 2. What is the Taylor series for about ? I.e., what is the MacLaurin series for ?
Exercise 3 [Math.SE]. Using Exercise 2, demonstrate using the MacLaurin series for that . You can work up to if you want.
Exercise 4 [Math.SE]. What is the MacLaurin series for the function ?
Exercise 5 [Math.SE]. Find the MacLaurin series for by first taking its derivative, then considering the MacLaurin series for this derivative, and then integrating the MacLaurin series.
Exercise 6 [Math.SE]. Using the MacLaurin series for sine and cosine, plug them into the formula , and what do we get?
Exercise 7 [Math.SE] What is the Taylor series for about ?
Exercise 8. Use the MacLaurin series for sine to prove
Exercise 9 [Math.SE]. What is the MacLaurin series for ?
Exercise 10. What is the Taylor series for about ? Or if you prefer, find the MacLaurin series for .
Exercise 11. What is the MacLaurin series for ?