Taylor Series Coefficients, Single Variable Calculus

If an infinitely differentiable function f(x) has a power series representation on the interval (a-R,a+R), then the coefficients for the power series are the Taylor coefficients: f^(n)(a)/n!, where f^(n)(a) represents the nth derivative of the function evaluated at the center x = a. Thus the Taylor series expansion of a function f(x) around the center x = a is given by: f(x) = f(a) + f'(a)(x - a)/1! + f''(a)(x - a)²/2! + f'''(a)(x - a)³/3! + ... We derive this form, discuss the radius of convergence, and work through an example. Watch next: https://youtu.be/u1GvZjd1JsQ Additional details: For the radius of convergence, we can employ an alternative to the ratio test. If the limit of the ratio of consecutive coefficients exists, the radius is the reciprocal of this limit. We conclude this lesson with a challenging example, finding the Taylor coefficients for 𝑓(𝑥)=1/√𝑥 centered at 𝑥=16. (This example is much more involving than the rest of our study of Taylor series!) To find the Taylor coefficients we need to compute the derivatives of 𝑓(𝑥), evaluate them at 𝑥=16, and then divide by 𝑛!. We differentiate to identify the pattern for the 𝑛-th derivative until we see that the 𝑛-th derivative will have a form that includes a product of odd numbers in the numerator and a power of 2 in the denominator. To find the radius of convergence, we can use the ratio test on the coefficients, finding R=16. The alternating series test and its remainder theorem give us the first three decimals for 1/sqrt(17) as 0.242. #mathematics #math #sequencesandseries #powerseries #taylorseries #calculus #iitjammathematics #maclaurinseries #ratiotest