# Tagged Questions

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### Closed-form term for this expression

I have a normal Distribution $X \sim N(\mu, \sigma)$. Is there an easy way to give an asymptotic estimate with small error (I would prefer with relative error $\rightarrow 0$) for $P[X \geq k]$? We ...
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### Asymptotics of a bounded function

We have given a function $f(x)$ where we know that $f(x)\leq 1$ for all $x$. Is it true that $$1+O(f(x))=O(f(x))$$ even though I know that $1 \geq f(x)$? We know that $O(f(x))=o(g(x))$. Is it true ...
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### Applications of the Exponential Integral?

this is my first time asking a question on here so please forgive me if I have made any formatting mistakes. I have the integral $f(x) = \int_0^\infty \frac{e^{-t}}{x + t} \; dt$ and I have shown the ...
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### Asymptotic approximation of a certain sum

During calculations of an expectation of some random variable, I have encountered the following sum: $$\sum_{t=2}^{n+1} \frac{t(t-1) \cdot n!}{(n-t+1)!\cdot n^t}$$ I ...
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### Why does the asymptotic equation of the modified Bessel of the second kind (Iv) have an imaginary part?

This is a follow up to this question. How does one arrive at the asymptotic expressions for the bessel functions? After looking at: G. N. Watson, "A Treatise on the Theory of Bessel Functions", 2nd ...
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### Asymptotics of sum of Binomial Coefficients (Binomial distribution) - Poisson approximation?

Let $$f(n):=\sum_{i=k}^n {n \choose i } p^i (1-p)^{n-i}$$ where $k\geq 2$ is a fixed Parameter and $p=p(n) \in (0,1]$ depends on $n$ where $np\leq 1$. We consider $n \rightarrow \infty$. I've found ...
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### How does one arrive at the asymptotic expressions for the bessel functions?

It is known that Bessel functions for large arguments will behave as exp or cos/sin however I was wondering how does one arrive at those results. The motivation being that I would like to use these ...
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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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### Roots of the equation $I_1(b x) - x I_0(b x) = 0$

I'm interested in the roots of the equation: $I_1(bx) - x I_0(bx) = 0$ Where $I_n(x)$ is the modified Bessel function of the first kind and $b$ is real positive constant. More specifically, I'm ...
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### “$O$” notation in Stirling approximation

In the Stirling approximation the formula as typically used in applications is $$\ln n! = n\ln n - n +O(\ln(n))$$ I'm confused about the last term "$O$" . What does it mean actually, and when do we ...
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### Method of dominant balance

Find the leading asymptotic behaviour as $x \rightarrow \infty$ of $$x^2y'' + (1 + 3x)y' + y = 0$$ Can someone kindly explain me how to solve this problem? Im learning asymptotic analysis, and I ...
I have a very messy function. It consists sums four levels deep, and the inner-most term is itself quite messy. $$\sum \sum \sum \sum (\mbox{stuff})$$ I can't find a closed form for this function. ...