 In this video, we'll find the Taylor series for natural log of x centered at x equals 1. We begin by finding several derivatives of the natural log of x. So the zeroth derivative is natural log of x, and if we evaluate that at 1, we get 0. The first derivative is 1 over x, evaluated at 1 gives us 1. The second derivative is negative 1 over x squared, evaluated at 1 is negative 1. The third derivative is 2 over x cubed, evaluated at 1 is 2. The fourth derivative is negative 2 times 3 over x to the fourth, evaluated at 1 is negative 6. And the fifth derivative is 2 times 3 times 4 divided by x to the fifth, evaluated at 1 gives us 24. So in generating this Taylor series for f of x being natural log of x centered at x equals 1, here's the general form evaluated at x equals 1. We generate a few terms. Times we have 0 divided by 0 factorial times x minus 1 to the 0, plus 1 over 1 factorial times x minus 1 to the first, minus 1 over 2 factorial times x minus 1 to the second, plus 2 over 3 factorial times x minus 1 to the third, minus 6 over 4 factorial times x minus 1 to the fourth, plus 24 over 5 factorial times x minus 1 to the fifth, minus, and so on. Of course, remembering that each of these coefficients, numerators, were generated in the slide before. Cleaning this up, we can write x minus 1 minus 1 over 2 times x minus 1 squared, plus 1 over 3 times x minus 1 cubed minus 1 over 4 times x minus 1 to the fourth, plus 1 over 5 times x minus 1 to the fifth, minus, and so on. And ultimately, we can write this in summation notation as the sum from k equals 1 to infinity of negative 1 to the k plus 1 over k times x minus 1 to the k. Verify this on your own paper.