Taylor expansions

31 problems · Taylor formulas and local expansions.

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Ex.10Expanding √x at x = 4

#Taylor expansion #shifted point #numerical estimate

Taylor expansions

Ex.32lim (1/x² − 1/sin² x)

#limits #expansions #difference of fractions

Taylor expansions

Ex.33lim ((1 + x)^(1/x) / e)^(1/x)

#limits #indeterminate form 1^∞ #logarithm #expansions

Taylor expansions

Ex.42lim cos(1/n)^(n²)

#sequences #expansions #indeterminate form 1^∞

Taylor expansions

Ex.56Expansions of 1/(1 + x + x²) and ln(1 + x + x²)

#Taylor expansion #geometric series #factorisation trick

Taylor expansions

Ex.62Position of a curve relative to its tangent

#tangent line #inflection #local behaviour

Taylor expansions

Ex.83How many terms are needed to compute sin 1?

#Taylor-Lagrange #error bound #numerical computation

Taylor expansions

Ex.579The binomial expansion of √(1 + x)

#Taylor expansion #binomial series #square root

Taylor expansions

Ex.580The Taylor expansion of arctan by integrating

#Taylor expansion #integration of expansions #arctangent

Taylor expansions

Ex.731Classifying a critical point when f″(0) = 0

#local extrema #Taylor expansion #second derivative test

Taylor expansions

Ex.778Approximating e with a guaranteed error

#Taylor-Lagrange #error bound #number e

Taylor expansions

Ex.796An equivalent of e^(sin x) − e^(tan x)

#equivalents #expansions #difference of functions

Taylor expansions

Ex.797A one-sided bound for √(1 + x) from the Lagrange remainder

#Taylor-Lagrange #inequality #remainder sign

Taylor expansions

Ex.798Computing ∫₀¹ sin(x²) dx with a series

#integration of expansions #numerical approximation #alternating series

Taylor expansions

Ex.840A fast series for ln 2

#logarithm #series #numerical approximation #speed of convergence

Taylor expansions

Ex.870cosh x ≤ e^(x²/2)

#series comparison #hyperbolic functions #inequalities

Taylor expansions

Ex.956Finite difference formulas for f′

#finite differences #numerical differentiation #error estimate

Taylor expansions

Ex.985Derivatives of high order at 0 without differentiating

#Taylor expansion #uniqueness #high order derivatives

Taylor expansions

Ex.1047Approximating x by a combination of sines

#undetermined coefficients #Taylor expansion #Richardson extrapolation

Taylor expansions

Ex.1175Convergence of three series via expansions

#series #equivalents #comparison

Taylor expansions

Ex.1181Expanding x/(eˣ − 1)

#Taylor expansion #quotient #Bernoulli numbers #undetermined coefficients

Taylor expansions

Ex.1182Taylor series of the solution of y′ = x + y²

#differential equation #Taylor expansion #successive derivatives

Taylor expansions

Ex.1183The symmetric second difference

#Taylor-Young #second derivative #counterexample

Taylor expansions

Ex.1523An expansion of order 2 without a second derivative

#Taylor-Young converse #counterexample #differentiability

Taylor expansions

Ex.1524A rational approximation of cosine

#Padé approximant #undetermined coefficients #approximation

Taylor expansions

Ex.1525Expanding the inverse of x + x³

#Taylor expansion #inverse function #fixed point iteration

Taylor expansions

Ex.1557Bounding f′ by f and f″ (Landau's inequality)

#Taylor-Lagrange #inequalities #optimisation of a parameter

Taylor expansions

Ex.1566Central binomial coefficients from (1 − 4x)^(−1/2)

#binomial series #generalised binomial coefficients #central binomial

Taylor expansions

Ex.1657e is irrational

#Taylor-Lagrange #irrationality #exponential

Taylor expansions

Ex.1883Asymptotic expansion of (1 + 1/n)ⁿ

#asymptotic expansion #sequences #composition of expansions

Taylor expansions

Ex.2083A smooth function that is not its Taylor series

#flat function #smooth #analytic #counterexample

Taylor expansions