 In this video, we'll find the Taylor series for cosine of x centered at x equals zero, otherwise known as the Maclaurin series for cosine of x. We begin by finding several derivatives of cosine of x. The zeroth derivative of cosine is cosine. Evaluated at zero gives us one. First derivative is negative sine of x. Evaluated at zero is zero. Second derivative is negative cosine of x. Evaluated at zero is negative one. Third derivative is sine of x. Evaluated at zero is zero. The fourth derivative is cosine of x. Evaluated at zero is one. So, in generating a few of the terms in this Taylor series, we have one divided by zero factorial x to the zero, plus zero divided by one factorial x to the first, minus one over two factorial x to the second. Notice I'm getting these coefficients from what we generated in the previous slide. Plus zero divided by three factorial x to the third, plus one over four factorial x to the fourth, and so on. We find that this is equal to one minus x squared over two, plus x to the fourth over four factorial, and so on. We notice that the terms are alternating and the powers are all even. So we can actually write this in general form as the sum from k equals zero to infinity of negative one to the 2k divided by 2k factorial x to the 2k. Verify this on your own paper.