Questions tagged [taylor-expansion]
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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Generator of translations in a Riemannian manifold
Let $f$ be a smooth function from $X=\mathbb{R}^n$ into $\mathbb{R}$. We have that
\begin{equation}
e^{\vec{\tau}\cdot \vec{\nabla}}f(\vec{x})=f(\vec{x}-\vec{\tau})
\end{equation}
where $\vec{\tau}\in ...
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Finding the Taylor Series for $f\left(x\right)=\frac{x}{e^{x}-1}$ [closed]
I want to see how one would figure out that $\frac{x}{e^{x}-1}=\sum_{n=0}^{\infty}B_{n}\frac{x^{n}}{n!}$ where $B_{n}$ represents the nth Bernoulli number with the convention that $B_{1}=-\frac{1}{2}$....
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Taylor's Theorem for functions on the Rational Numbers
I have been looking for different proofs of the Taylor's Theorem with Peano form of the remainder. But all proofs I have found use some form of the Mean Value Theorem or L'Hôpital's Rule (which also ...
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Taylor series expnasion
I was given the following question :
Evaluate the series expansion of $f(x) = \ln(\frac{1-x}{1+x^2})$ and determine the radius of convergence.
So my first move was to split the logarithm into $\ln(1-x)...
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Solving a combined limit with an $1^{\infty}$ form nested inside a 0×∞ form
I came across this limit problem:
$\lim _{x \rightarrow \infty}\left\{\left(\frac{x+1}{x-1}\right)^x-e^2\right\} \cdot x^2$
Plugging this into desmos, one can see that the limit approaches $\frac{2 e^...
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'deducing' a bound using the first order taylor series. How to make it more precise?
So, I just saw a ‘proof’ that the generalized birthday problem has a median of C*sqrt(n). Though the probability in question is interesting, this question is more about calculus and maybe asymptotics ...
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Construct/prove existence of a function with given expansions at two different points
Consider two non-constant real polynomials $f(x)$ and $g(x)$:
$$f=f_0 + f_1 (x-x_0) +...+f_N(x-x_0)^N $$
$$g=g_0 + g_1(x-x_1) +...+g_M(x-x_1)^M $$
where $f_0...f_N,g_0...g_M,x,x_0,x_1 \in \mathbb{R}$ ...
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How to show that $1-\sqrt{\dfrac{2}{n}}\dfrac{\Gamma(\frac{n+1}{2})}{\Gamma(\frac{n}{2})}<\dfrac{1}{4n}$
In the calculation of a problem, I need to show that $1-\sqrt{\dfrac{2}{n}}\dfrac{\Gamma(\frac{n+1}{2})}{\Gamma(\frac{n}{2})}<\dfrac{1}{4n}$ holds for any positive integer $n$. I got the expansion ...
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Taylor expansion of $\int_{0}^{\omega_0}\frac{\sin\left(\frac{2N+1}{2}\omega\right)\cos(\omega n)}{\sin\left(\omega/2\right)}d\omega$ and similar
I would like to compute the expansion of the following integrals near $N = + \infty$ up to $\mathcal{O}(1/N^2)$: $$ \int_{0}^{\omega_0} \frac{\sin \left( \frac{2 N + 1}{2} \omega \right) \cos(\omega n)...
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How do we determine what small angle and small $x$ are for a simple pendulum to justify linear approximation?
Consider a simple pendulum consisting of a point-like mass $m$ attached to a massless string of length $L$ from a fixed support and constrained to move in a vertical plane.
Here is a picture of this
...
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$\sin(x) \leq \sum_{k=0}^n (-1)^k \frac{x^{2k+1}}{(2k+1)!}$ when $n$ is even and $x\geq 0$
Let $n\geq 2$ be an even integer and $x\geq 0$. I want to show that
$$\sin(x) \leq \sum_{k=0}^n (-1)^k \frac{x^{2k+1}}{(2k+1)!}.$$
Assume first that $x\in (0,\pi]$.
By Taylor's theorem with Lagrange ...
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Do you need L'Hôpital's rule to prove Taylor's formula?
I recently read a Quora answer. The answerer was asked to solve the limit
$$\lim_{x\to0}\frac{\cos x-e^x}{\sin x}$$
without using L'Hôpital's rule. The answerer used the Taylor series expansion of the ...
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Find the Laurent expansion of $f(z) = \sin{\frac{1}{z(z-1)}}$ in $0<|z-1|<1$
Here is my idea:
$\sin{\frac{1}{z(z-1)}} = \sin{\left( \frac{-1}{z} + \frac{1}{z-1}\right)} = \sin{\left(\frac{1}{z-1} - \frac{1}{z}\right)} = \sin{\frac{1}{z-1}}\cos{\frac{1}{z}} - \cos{\frac{1}{z-1}}...
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summation of a series on a lattice
I am a physics student. And I am working on some wave function on some lattice. One question I encountered in my study is studying the following summation
$$f(x)=\sum_{m,n \in Z}\cos((m-n)gx)\exp(-\...
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Prove $\ln (1+x) \leq x - x^2/4 $ for $x \leq 1$ using Taylor's theorem
RTP: $\ln (1+x) \leq x - \frac{1}{4} x^2$ for $x \leq 1$
I am trying to prove this specifically using Taylor theorem. Here is what I have so far:
$\ln (1+x) = x - \frac{x^2}{2} + \frac{x^3}{3(1+\xi)^3}...
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