Tagged Questions

Questions regarding the Taylor series expansion of univariate and multivariate functions, including coefficients and bounds on remainders. A special case is also known as the Maclaurin series.

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Will values assigned to divergent series match a taylor series past the radius of convergence?

With what I've seen in nearly every case this is true but there are some cases where the function goes to infinity. I'm thinking specifically $y=ln(x-1)$, $y=1/(x-1)$, and $y=(x-1)^2$ centered at 0 ...
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Remainder of Taylor approximation

Consider the ODE $\dot{x}=f(x)$ with $f(x)$ smooth and let $x_0$ be an equilibrium, i.e. $x(t)=x_0=\text{const}$ and $f(x_0)=0$. The substitution $x=x_0+y$ shifts the origin to $x_0$. With the new ...
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A funny question: Taylor polynomials and series associated with the Lost numbers $4, 8, 15, 16, 23, 42$

The interpolation polynomial for the "Lost" numbers $4, 8, 15, 16, 23, 42$ is $$P(x)=60-\frac{612}{5}x+\frac{367}{4}x^{2}-\frac{235}{8}x^{3}+\frac{17}{4}x^{4}-\frac{9}{40}x^{5}.$$ That is, $P(1)=4$, ...
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Series expansion for $x$, when $x$ is small

Suppose that we are given the series expansion of $y$ in terms of $x$, where $|x|\ll 1$. For example, consider $$y=x+x^2+x^3+\cdots\qquad\qquad\qquad (1).$$ From this I would like to derive the series ...
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What could the ratio of two sides of a triangle possibly have to do with exponential functions?

Name says it all. The two seem so unrelated? What's more, if I'm not mistaken the exponential version contains an imaginary part. I'm kind of ignorant about imaginary numbers, but does this mean that ...
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Log-linearizing $Y_t=\int_0^1 F(X_{it}) di$

I want to prove that log-linearizing the expression $Y_t=\int_0^1 F(X_{it}) di$ yields: $$Yy_t \approx F'(X)X\int_0^1 x_{it} di$$ Where: $\{X_{it}\}_{i \in (0,1)}$ is a continuum of strictly ...
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I would like to prove that : $$(1+\ldots+o(x^{n-1}))^4=(1+\ldots+o(x^{n-1}))$$ i took that from the picture below My Proof: note that $$(1+x)^a = 1 + ax + \frac{a(a-1)}{2!}x^2 + \frac{a(a-... 1answer 43 views Differentials squared - Divergence in general orthogonal curvilinear coordinates. I was reading this document on how to get some common operators when dealing with general orthogonal curvilinear coordinates. I am interested in particular in equation (12). It basically defines three ... 2answers 30 views \exists C>0 such that \frac{1}{2}(e^x+e^{-x}) \le e^{\frac{1}{2}x^2-Cx}? It is well-known and easy to check that for any real x it holds$$ \frac{1}{2}(e^x+e^{-x}) \le e^{\frac{1}{2}x^2}.  [To show this, it is sufficient to write explicitly their Taylor series ...
Consider the function dependent on the variables $N_t$ and $N_{t-1}$. Call the function $f$ so $f = f(N_t, N_{t-1})$. Now suppose we could write $N_t = N^*+n_t$ where $N^*$ is constant, and $n_t$ ...
We have a function $V(x)$ (potential energy) with $x$ being some variable. This function has a minimum at a certain $x_0$. We assume that $V(x)$ is an analytic real function of $x$ around $x_0$. ...