# Tagged Questions

Questions involving asymptotic analysis, including growth of functions, Big-O notation, Big-Omega and Big-Theta notations.

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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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### On $\sum_{\substack{\zeta(\frac{1}{2}+i\gamma)=0\\0<\gamma<T}}\prod_{n=1}^\infty \left| 1-\frac{(\gamma\log x)^2}{n^2\pi^2}\right|$ as $O(\log x)$

On assumption that the identity (2) for a representation of $\pi(x)$ holds, see here Two Representations of the Prime Counting Function in this site Mathematics Stack Exchange, and since using the ...
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### Show that $\frac{(-1)^{n}}{\left(\ln(n)+(-1)^{n}\right)^{2}}=\frac{(-1)^{n}}{\ln^{2}(n)}+v_n\quad \left( v_n\sim -\frac{2}{\ln^{3}(n)} \right)$

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

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

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

The bound for $$\sum_{i=1}^n\binom{n}{i}2^i$$ is $O\left(3^n\right)$ but what will be the bound for $$\sum_{i=1}^{\frac{n}{2}}\binom{n}{i}2^i$$ Any idea how should I proceed?
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### Bound on binomial summation

The bound for $\sum_{i=1}^n\binom{n}{i}2^i$ is $O(3^n)$ but what will be the bound for $\sum_{i=1}^{\frac{n}{2}}\binom{n}{i}2^i$. Any idea how should I proceed.
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### multi-scale analysis integral constraint

If I have a function $u(x)$ and I am interested in multiscale analysis for $\tilde{u}(\tilde{x},X)$ where $\tilde{x}=x,X=\epsilon x$, in the context of solving some PDE- the differential operators are ...
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### CLRS substitution method “subtracting constant” technique

I'm reading CLRS, and in Chapter 4 it states that if you guess the asymptotic complexity of a recurrence correctly but cannot quite get the mathematical induction work out, a common method to employ ...
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### Equivalent of $\int_2^{+ \infty} e^{\Gamma (t) \log x}dt$ when $x \to 1$

I wonder if the equivalent : $$\int_2^{+ \infty} e^{\Gamma (t) \log x}dt$$ for $x \to 1^{-}$ (i.e the first term in the asymptotic expansion) had been studied ? Is it tricky to get an equivalent ?...
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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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### What is the value of $I=\lim_{n \to \infty} \int_0^1 {{1 + nx^2}\over{(1 + x^2)^n}} \log(2 + \cos(x/n))\,dx.$?

Find the integral $I$.....it looks like a good problem which I was not able to solve ....please help... $$I=\lim_{n \to \infty} \int_0^1 {{1 + nx^2}\over{(1 + x^2)^n}} \log(2 + \cos(x/n))\,dx.$$
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### Show : $(-1)^{n}n^{-\tan\left(\tfrac{\pi}{4}+\tfrac{1}{n} \right)}=\tfrac{(-1)^{n}}{n}+\mathcal{O}\left(\tfrac{\ln(n)}{n^{2}} \right)$

I would like to show that : $$(-1)^{n}n^{-\tan\left(\dfrac{\pi}{4}+\dfrac{1}{n} \right)}=\dfrac{(-1)^{n}}{n}+\mathcal{O}\left(\dfrac{\ln(n)}{n^{2}} \right)$$ My proof: Note that : \begin{...
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### Show that $\tfrac{(-1)^{n}}{\cos(n)+n^{\tfrac{3}{4}}}=\tfrac{(-1)^{n}}{n^{\tfrac{3}{4}}}+\mathcal{O}\left( \tfrac{1}{n^{\tfrac{3}{2}}}\right)$

I would like to show that : $$\dfrac{(-1)^{n}}{\cos(n)+n^{\tfrac{3}{4}}}=\dfrac{(-1)^{n}}{n^{\tfrac{3}{4}}}+\mathcal{O}\left( \dfrac{1}{n^{\tfrac{3}{2}}}\right)$$ by starting from the left side ...
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### Show that $(-1)^{n}\sqrt[n]{n}\sin(\frac{1}{n})=\tfrac{(-1)^{n}}{n}+\mathcal{O}\left(\tfrac{\ln(n)}{n^{2}} \right)$

I would like to show that : $$(-1)^{n}\sqrt[n]{n}\sin(\frac{1}{n})=\dfrac{(-1)^{n}}{n}+\mathcal{O}\left(\dfrac{\ln(n)}{n^{2}} \right)$$ by starting from the left side and get the right side : My ...
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### $X-x_0=O_p(n^{-1/2})$ implies $g(X)-g(x_0)=O_p(n^{-1/2})$

Suppose that $X$ is a random vector and $x_0$ is a fixed vector such that $$X-x_0=O_p(n^{-1/2}).\tag{*}$$ Let $Y=g(X)$ where $g$ has a continuous gradient that is nonzero at $x_0$. Let $y_0=g(x_0)$...
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### Show that $(-1)^{n}\left(\sqrt{n+1}-\sqrt{n} \right)=\tfrac{(-1)^{n}}{2\sqrt{n}}+\mathcal{O}\left(\tfrac{1}{n^{\frac{3}{2}}} \right)$

I would like to show that : $$\fbox{(-1)^{n}\left(\sqrt{n+1}-\sqrt{n} \right)=\dfrac{(-1)^{n}}{2\sqrt{n}}+\mathcal{O}\left(\dfrac{1}{n^{\dfrac{3}{2}}} \right)}$$ by starting from the left side ...
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### Evaluating a polynomial that includes a little-o constant exponent

In a paper I am reading, the following assertion is made: N is an odd, positive composite number. Let $I_1 = [\sqrt N - F^{2+o(1)} , \sqrt N + F^{2+o(1)}]$ hold the values of $x$ passed into the ...
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### Difference/switch between big/small o in taylor series

for example i only know taylor series with small o is there anyway to switch from small o to big o in taylor series and why when we want to see the nature of some series we use taylor series with ...
For $n\in\mathbb N^*$, we consider the triangular matrix $$T_n = \begin{pmatrix} 1 & \cdots & 1 \\ & \ddots & \vdots \\ 0 & & 1 \end{pmatrix} \in M_{n,n}(\mathbb R) \,.$$ ...