What is the radius of convergence for Sinx?
sinxx=∞∑n=0(−1)nx2n(2n+1)! with radius of convergence R=∞ .
What is power series expansion of Sinx?
Theorem. The sine function has the power series expansion: sinx. = ∞∑n=0(−1)nx2n+1(2n+1)!
How do you write Sinx in exponential form?
sinx=x−x33!
What is the Taylor remainder theorem?
Taylor’s Formula: If f(x) has derivatives of all orders in a n open interval I containing a, then for each positive integer n and for each x ∈ I, f(x) = f(a) + f (a)(x − a) + f (a) 2!
What does Taylor’s formula for sin ( x ) tell us?
Taylor’s formula now tells us that: sin(x) = 0 + 1x + 0x2 + 4 3! 1 x 3 + 0x + ··· x3 5 7 = x − 3! 5! 7! Notice that the signs alternate and the denominators get very big; factorials grow very fast. The radius of convergence R is infinity; let’s see why.
What kind of approximation does Taylor’s theorem give?
In calculus, Taylor’s theorem gives an approximation of a k -times differentiable function around a given point by a polynomial of degree k, called the k th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order k of the Taylor series of the function.
How is Taylor’s theorem generalized to multivariate and vector valued functions?
Taylor’s theorem also generalizes to multivariate and vector valued functions. Graph of f(x) = ex (blue) with its linear approximation P1(x) = 1 + x (red) at a = 0.
How is the exponential function y related to Taylor’s theorem?
The exponential function y = e x (red) and the corresponding Taylor polynomial of degree four (dashed green) around the origin. Part of a series of articles about. Calculus. In calculus, Taylor’s theorem gives an approximation of a k-times differentiable function around a given point by a k-th order Taylor polynomial.