How do you find the Taylor series for sin x centered at pi?
Recall that the formula for the Taylor series of the function f centered at
So let’s figure out the first few terms and see if we notice a pattern. For that, we’ll need the derivatives of their evaluations at x=π
We can easily see that it repeats from there. So the even derivatives of sin(x) at x=π are all zero and the odd derivatives alternate between −1 and 1. Therefore, we see that an expression for the derivatives is given by
Plugging in, we get
It would be nice if we didn’t have to explicitly write the “k odd” part below the summation, though. Note that k′=2k−1 gives us all of the odd numbers only. So let’s replace k everywhere in the expression with 2k−1 to get
That final expression then is our Taylor series.