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In calculus, Taylor's theorem gives an approximation of a -times differentiable function around a given point by a polynomial of degree , called the -th-order Taylor polynomial. For a smooth function , the Taylor polynomial is the truncation at the order k {\textstyle k} of the Taylor series of the function.
That is, the Taylor series diverges at x if the distance between x and b is larger than the radius of convergence. The Taylor series can be used to calculate the value of an entire function at every point, if the value of the function, and of all of its derivatives, are known at a single point. Uses of the Taylor series for analytic functions ...
Real analysis is an area of analysis that studies concepts such as sequences and their limits, continuity, differentiation, integration and sequences of functions. By definition, real analysis focuses on the real numbers, often including positive and negative infinity to form the extended real line.
Taylor's theorem – gives an approximation of a times differentiable function around a given point by a -th order Taylor-polynomial. L'Hôpital's rule – uses derivatives to help evaluate limits involving indeterminate forms; Abel's theorem – relates the limit of a power series to the sum of its coefficients
Pages in category "Theorems in real analysis" The following 45 pages are in this category, out of 45 total. ... Taylor's theorem; Titchmarsh convolution theorem; U.
Extreme value theorem; Differential equation; Differential operator; Newton's method; Taylor's theorem; L'Hôpital's rule; General Leibniz rule; Mean value theorem; Logarithmic derivative; Differential (calculus) Related rates; Regiomontanus' angle maximization problem; Rolle's theorem
Given a twice continuously differentiable function of one real variable, Taylor's theorem for the case = states that = + ′ () + where is the remainder term. The linear approximation is obtained by dropping the remainder: () + ′ ().
Danskin's theorem (convex analysis) Darboux's theorem (real analysis) Darboux's theorem (symplectic topology) Davenport–Schmidt theorem (number theory, Diophantine approximations) Dawson–Gärtner theorem (asymptotic analysis) de Branges's theorem (complex analysis) de Bruijn's theorem (discrete geometry) De Bruijn–Erdős theorem ...