Taylor's Theorem with four proofs (Peano, Integral form, Cauchy, Lagrange), Real Analysis

In this lesson, we examine Taylor's theorem, presenting and rigorously proving four of its primary variants: Peano's form of the remainder, the integral form, Cauchy's form, and Lagrange's form. Each of these forms provides a distinct characterization of the remainder term, which is crucial for understanding the error in Taylor polynomial approximations. We begin with a brief review of Taylor series, then proceed to the proofs. 1) Peano's form characterizes the remainder as a function that vanishes faster than (x - a)^k as x approaches a, where k is the degree of the Taylor polynomial. This provides a qualitative understanding of the remainder's behavior near the point of expansion. 2) The integral form expresses the remainder as a definite integral involving the (k+1)-th derivative of the function. 3) Cauchy's form expresses the remainder as a product involving the (k+1)-th derivative evaluated at an intermediate point. 4) Lagrange's form also involves the (k+1)-th derivative at an intermediate point, and puts the remainder in a form that resembles the Taylor series terms most closely. We also discuss the necessary conditions for these theorems, emphasizing the roles of differentiability and continuity in the hypotheses. Review Taylor Series here: • Coefficients (https://youtu.be/G6lnFAMS0w0) • Taylor series for ln(1+x) (https://youtu.be/u1GvZjd1JsQ) • Taylor series for e^x (https://youtu.be/Wryqq3fzQWc) • Taylor series for sin(x) and cos(x) (https://youtu.be/Btuhq92xYBo) • arctan(x) (https://youtu.be/JreVcerP5oE) #TaylorsTheorem #realanalysis #taylorseries #advancedcalculus #mathematicalanalysis #differentialcalculus #proof #mathematics #maths