# 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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### Show $\sin\left(2\pi\sqrt{n^2+(-1)^{n}} \right)=\frac{(-1)^{n}\pi}{n}+\mathcal{O}\left( \frac{1}{n^2}\right)$

I would like to prove the following: $$\sin\left(2\pi\sqrt{n^2+(-1)^{n}} \right)=\dfrac{(-1)^{n}\pi}{n}+\mathcal{O}\left( \dfrac{1}{n^2}\right).$$ My attempt: \begin{align*} \sin\left(2\pi\sqrt{n^...
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### Show $\cos\left( \pi n^{2}\ln\left(\frac{n}{n-1} \right) \right)=(-1)^{n+1}\frac{\pi}{3n}+\mathcal{O}\left( \frac{1}{n^2}\right)$

I would like to show : $$\cos\left( \pi n^{2}\ln\left(\dfrac{n}{n-1} \right) \right)=(-1)^{n+1}\dfrac{\pi}{3n}+\mathcal{O}\left( \dfrac{1}{n^2}\right)$$ by starting from the left side and get the ...
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### Show that $(-1)^{n}\left( (n+1)^{\frac{1}{n+1}}-n^{\frac{1}{n}}\right)=\mathcal{O}\left(\frac{\ln(n)}{n} \right)$

I would like to show: $$(-1)^{n}\left( (n+1)^{\dfrac{1}{n+1}}-n^{\dfrac{1}{n}}\right)=\mathcal{O}\left(\dfrac{\ln(n)}{n} \right)$$ Here is my attempt \begin{align*} (-1)^{n}\left( (n+1)^{\dfrac{1}{...
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### Baker-Campbell-Hausdorff/Zassenhaus formula to first order in one matrix

Is there a closed-form expression for the term of $e^{t(c \hat{X} + d \hat{Y})}$ that is first-order in $d$, where $t$, $c$, and $d$ are scalars and $\hat{X}$ and $\hat{Y}$ are finite-dimensional ...
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### Solving a nonlinear equation $\sum_{z=0}^{s} \frac{(\lambda(l-x))^z}{z!} e^{-\lambda(l-x)}=p$

I would appreciate it if someone helps me with solving the following equation. Suppose $\lambda,l \in R^+$, $p\in[0,1]$, and $s\in N_{0}$. How can we find an $x\in [0,l]$, which satisfies the ...
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### Upper and lower bound for maclaurin series of exponential function [on hold]

I have an algorithm like this: The algorithm and I want to find upper bound for O() notation and lower bound for Ω() notation. When I try debug the algorithm, It is maclaurin series but without 1,...
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### Lagrange remainder vs. Alternating Series Estimation Theorem: do they always give you the same error bound?

Given a function and its nth degree Taylor series approximation, we can use the Lagrange form of the remainder to get a maximum value of the error of approximation. If the series is also an ...
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### Asymptotic expansion of elliptic integral

I am trying to find the first 2-3 terms of the asymptotic expansion in terms of 1/ρ of the elliptic integral I_n(\rho)=\int_0^\frac{h_2}{\rho}\frac{t^{2n}/h_2^{2n}}{(E_n(t))^2\...
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Is there a formula to multiply many series (More than two) using the Cauchy product? If there isn't, please tell me how I can write this formula $\left( \frac{1}{a-e^x} \right) ^{n+1}$as the ...
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### Prove that this limit is the logarithmic derivative of the Riemann zeta function.

Prove the following limit: $$-\frac{\zeta '(s)}{\zeta (s)}=\lim_{c\to 1} \, \left(\frac{\zeta (c) \zeta (s)}{\zeta (c+s-1)}-\zeta (c)\right)$$ As a starting point I tried to enter this series ...
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### Convergence of the series $\sum \frac{(-1)^{n}}{n^{\frac{2}{3}}+n^{\frac{1}{3}}+(-1)^{n}}$

To prove that nature of the following series : $$\sum \dfrac{(-1)^{n}}{n^{\frac{2}{3}}+n^{\frac{1}{3}}+(-1)^{n}}$$ they use in solution manual : My questions: I don't know how to achieve ( * ) ...
### Can the series $\sum_{n=0}^\infty \frac{(-1)^n}{\sqrt{2^n n!}} x^{n}$ be summed? [on hold]
Can the following series $$\sum_{n=0}^\infty \frac{(-1)^n}{\sqrt{2^n n!}} x^{n}$$ be summed?