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How to prove there is not a polynomial $P(x)$ that is a good approximation to $\cos(x)$ for all $x$?

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#functions #trigonometry #proof-explanation #taylor-expansion #approximation
128
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Show series $\sum_{k=1}^{\infty} \frac{(2k-2) !\left(\frac{1}{4}\right)^{k-1}\left(1-\frac{1}{4}\right)^k}{k !(k-1) !}=1.$

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#calculus #integration #sequences-and-series #analysis #taylor-expansion
14
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prove that $|\log(1 + \frac{n-x}{x})| \geq \frac{|h|}{2x}$ where $n = [x] + h$ and $3 \leq |h| \leq \frac{x}{4}$

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#calculus #taylor-expansion
121
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What is the function whose Maclaurin series is given by $1 + \frac{3}{4} \, x^2 + \frac{75}{128} \, x^4 + \frac{245}{512} \, x^6 + \dots$?

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#calculus #sequences-and-series #taylor-expansion #power-series #special-functions
78
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Can a combination of EllipticK and EllipticE have a series expansion of the form $f(x)=1+\frac{5}{12}x^2+\frac{35}{128}x^4+\frac{105}{512}x^6+\dots$?

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#real-analysis #calculus #sequences-and-series #taylor-expansion #power-series
39
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Laurent series of real part of holomorphic function

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#complex-analysis #taylor-expansion #laurent-series
26
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Can a second-order expansion without the linear term be meaningful to estimate out-of-equilibrium oscillations?

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#taylor-expansion #dynamical-systems #mathematical-physics #quantum-mechanics
48
Views

Decision procedure for whether the power series of a rational function has only nonnegative coefficients

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#taylor-expansion #power-series #rational-functions #decision-problems
130
Views

Maximum and Minimum of $\sin x + \cos x$ using Taylor series

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#sequences-and-series #trigonometry #inequality #taylor-expansion
53
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A smooth, non-analytic real function which is not flat?

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#real-analysis #taylor-expansion #analyticity
16
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Question about multidimensional polynomials basis develoment

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#taylor-expansion #orthogonal-polynomials #hermite-polynomials
95
Views

Finding the function with power series $\sum _{n=0}^{\infty }\frac{x^{4n}}{(2n)!} $

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#calculus #sequences-and-series #taylor-expansion #power-series
185
Views

What does it mean by taylor series of a function about a point?

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#taylor-expansion
28
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Error analysis of a numerical time-averaging via Poisson equation

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#inequality #numerical-methods #taylor-expansion #poissons-equation
64
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Finding out where a complex function is holomorphic and relating this to its expansion as a power series.

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#sequences-and-series #complex-analysis #convergence-divergence #taylor-expansion #uniform-convergence

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