# Tagged Questions

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### Can I prove this, or hopeless? Deviating too much from mean

Can I prove this: We have a sequence of vectors $\left(X_i(n)\right)$ for $i=1,\ldots,t$, where $n\rightarrow \infty$. $t$ does depend on $n$ and is Chosen such that $1 \ll t \ll n$, for instance, ...
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### Properties of Asymptotic series Expansion

I am wondering about the properties of "Asymptotic series expansion". Considering a representative function $f(R)=\frac{a+bR+cR^2}{d+eR+fR^2}$ where $a, b, c , d , e , f$ are constants. How ...
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### What is the proper way to handle the limit with little-$o$?

I was hoping to show that $$\left(1-\frac{x}{n}+o\left(\frac{2x}{n}\right)\right)^n \xrightarrow{n\to\infty} e^{-x}$$ which would be just fine without the little-$o$. Trying binomial formula: ...
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### Understanding Recurrence Relation

as i ask question and answered by some Clever people at this topic: Recurrence Relation Solving Problem i try to learn new thing with new question very similar to get familiar with recurrence ...
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### Summation of $1/(xy)$ over a triangular region

Let us consider a lattice formed by all points with integer and positive coordinates on a Cartesian plane, and where $K$ is the maximal value for the x-axis. Let us assign to each lattice point the ...
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### Convergence of power towers

Let's define the sequence $\{s_n\}$ recursively as $$s_1=\sqrt2,\ \ \ s_{n+1}=\sqrt2^{\,s_n}.$$ Or, in other words, $$s_n=\underbrace{\sqrt2^{\sqrt2^{\ .^{\ .^{\ .^{\sqrt2}}}}}}_{n\ \text{levels}}.$$ ...
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### Infinite Series of the asymptotic expansion of Fresnel Integrals

I need to find the infinite series for the asymptotic expansions of the fresnel integrals as $x\rightarrow \infty$ and $x\rightarrow 0$. Now I have computed the asyptotic expansions to be as follows ...
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### How prove this sequence $a_{n}=\sqrt{n}+\frac{1}{2}-\frac{1}{8\sqrt{n}}+o\left(\frac{1}{\sqrt{n}}\right)?$

let sequence $\{a_{n}\}$ such $$a_{1}=1,a_{n+1}=1+\dfrac{n}{a_{n}}$$ show that: $$a_{n}=\sqrt{n}+\dfrac{1}{2}-\dfrac{1}{8\sqrt{n}}+o\left(\dfrac{1}{\sqrt{n}}\right)?$$ This result is china student ...
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### asymptotics of this sum $x \to 0$

given the sum $$\sum_{n=0}^\infty \frac{\exp(-nx)}{n+a} =f(x)$$ what would be the asymtptic of this series ?? for $a=1$ i believe this series goes as $f(x) \sim \frac{1}{x}+ \gamma$ for every ...
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### Asymptotics of sequence depending on Tricomi's function

I'm dealing with the following sequence $$p_n = U(a, a - n, 1)$$ where $a > 0$ and $U$ is Tricomi's function. I suspect that asymptotically when $n \to \infty$ its behaviour is a power law ...
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### Is there a “natural” subsequence of positive integers $k_1 < k_2 < \ldots$ such that $\sum_{i=1}^n \frac{1}{k_i} = \Theta (\log \log \log n)$?
The harmonic series partial sums grow like $\log n$, and the sum of inverses of the first $n$ primes grows like $\log \log n$. Is there an example of a "nautral" subset of the positive integers (say ...
### The asymptotic expansion for the weighted sum of divisors $\sum_{n\leq x} \frac{d(n)}{n}$
I am trying to solve a problem about the divisor function. Let us call $d(n)$ the classical divisor function, i.e. $d(n)=\sum_{d|n}$ is the number of divisors of the integer $n$. It is well known that ...