# Tagged Questions

For questions that involve concrete approximations, such as finding an approximate value of a number with some precision. For questions that belong to the mathematical area of Approximation Theory, use (approximation-theory).

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### Approximating on a line

Say I have sampled some points in $[0,1]^2$ and evaluate a function $f(x,y)$ for them. I am interested in the behavior of $f$ along a single dimension. If the points were like ...
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### How do you Taylor expand the log likelihood function of the Poisson distribution?

This question is an extension to this previous question asked by myself: When doing a maximum likelihood fit, we often take a ‘Gaussian approximation’. This problem works through the case of a ...
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### For $\sum_{n=1}^\infty (-1)^n\frac{(x-3)^n}n$ approximate when x = 4, with an error that does not exceed .01

For $\sum_{n=1}^\infty (-1)^n\frac{(x-3)^n}n$ approximate with specific details the series when x = 4, but with an error that does not exceed .01. That is, find a value of n so that the nth partial ...
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### Approximating $|1-e^{i\delta}|$

Actually this question is derived from the Rudin's Real and Complex Analysis's lemma 15.17. In the lemma, given condition is below; Let $h(z) \in H(\Omega)$ such that Re $h(z) = \log |1-z|$, |Im ...
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### Taylor polynomial approximation with absolute error less than 3 decimal places

Assume that f is a function with $|f^{(n)}(x)| \le 11,$ for all n and all real x. Let $T_n(x)$ denote the Taylor polynomial of degree n for f(x) about the point $x=0$. What is the least integer n ...
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### Can I get an approximation for $(1-x)^n$, where $0<x<1$, $n\gg 1$?

I know it can be done when $xn \ll 1$, but what about the cases when $xn \gt 1$ ? My best try is to use sth like: \begin{align*} (1-x)^n &= \sum\limits_{j=0}^{\infty}\left( \begin{array}{c} n ...
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### Is there any correlation between approximation trendline parameters?

Let's say I have two data sets $(x,y)$ and $(p,q)$ and two approximation trendlines: Logarithmic: $y = b·ln(x) + a$ Linear: $y = bx + a$ Let's say I applied logarithmic approximation to both data ...
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### Computation of a sum using Stirling's approximation and Watson's lemma

$$Ω=\sum_{n=0}^{N-\frac{E}{\epsilon}} \frac{Ν!}{\left(\frac{N-n-\frac{E}{\epsilon}}{2}\right)!\left(\frac{N-n+\frac{E}{\epsilon}}{2}\right)!n!}$$ I am supposed to calculate the above sum using first ...
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### Why does this expression equal pi?

I was fiddling with numbers when I noticed that $$50 \times 1.05^{168} \times \frac{12600}{727767941} \approx \pi$$ I understand it's an approximation. Does anyone know why?
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### asymptotic expansion/approximation

Find the small solution of $$y''-y\left ( 1-y^{2} \right )=0 \text{ with } y\left ( 0 \right )=\epsilon \ll 1$$ Making a pun, I decided that $$y^{3}\left ( 0 \right )\ll y\left ( 0 \right )$$ so ...
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### Uniform error of approximating the Heaviside function by a partial sum of its Fourier series

Suppose $f(x)=H(x-.5)$ where H = Heaviside function on $0<x<1$ is approximated by the first five nonzero terms of its Fourier sine series. Compute the uniform error (i.e maximum error, max p(x) ...
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### How to treat small number within square root

guys.I am reading a math book. It has a equation shown as follows, $\sqrt{(1+\Delta^2)}$ And then,since $\Delta$ is very small, it can be written as, $\sqrt{(1+\Delta^2)} = (1+\frac12\Delta^2)$ ...
### Prove that if $\mathbb{C}-K$ is connected, then $K$ is polynomially convex.
Prove that if $\mathbb{C}-K$ is connected, then $K$ is polynomially convex. $K$ polynomially convex means $K=\hat{K}$ where \hat{K}=\{z\in\mathbb{C}:|p(z)|\le \max_{ζ\in K}|p(ζ)|\text{ for all ...
### Let $f(x)=1/x$ and prove that $f[x_0,x_1,…,x_n]=\prod_{i=0}^nx_i^{-1}$. [closed]
Let $f(x)=1/x$ and prove that $f[x_0,x_1,...,x_n]=\prod_{i=0}^nx_i^{-1}$. I'm sure how to approach this or even how/why we need $f(x)=1/x$. Any solutions or hints are greatly appreciated.