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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1answer
17 views

Asymptotic approximation to radial wave function

This seemingly easy analysis is driving me up the wall. $$\frac{\text{d}^{2}u}{\text{d}\rho^{2}}=\left[1-\frac{\rho_{0}}{\rho}+\frac{l(l+1)}{\rho^{2}}\right]u$$ why is it for $$\rho\rightarrow0,$$ ...
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0answers
18 views

Softmax for Continuous Functions?

The softmax $\log \sum_{i=1}^n \exp(f_i)$ of vector $f$ is a smooth upper bound on $\max_i f_i$. However, the same cannot be said of $\log \int_{X} \exp(f(x))dx$ in relation to $\max_{x \in X} f(x)$ ...
1
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2answers
31 views

Approximation of $\frac{x}{\sqrt{x^2+R^2}}$

How do you prove this statement? If $x\gg R$ then $$\frac{x}{\sqrt{x^2+R^2}}\cong 1-\frac{1}{2}\left(\frac{R}{x}\right)^2$$ I have no ideas even how to start.
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1answer
38 views

Method for finding square roots quickly (manual)

I was recently studying AC circuits and there I need to use Pythagoras theorem a lot.So I was looking for a method with which square roots can be calculated very fast,manually up to 1 decimal ...
2
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1answer
62 views

Find a>1 s.t. $a^x = x$ has a unique solution

What $a$ makes $\{x\mid a^x = x\}$ a singleton? $$(1.4444)^x - x \le 0 \tag 1$$ has real solutions. $$(1.4447)^x - x \le 0 \tag 2$$ has no real solutions. I guess $1.4444 < a < 1.4447$ I ...
1
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0answers
25 views

Asymptotic behaviour of generalized binomial coefficient $\frac{an(an-1)…(an-n+1)}{n!}$

Let $a\in(0,1)$. What is the asymptotic behaviour of $\frac{an(an-1)...(an-n+1)}{n!}$ as $n\rightarrow\infty$? It looks like it might be possible to express this in terms of gamma functions and use ...
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1answer
33 views

Order of the error for the Trapezoidal and Simpson's method of numerical integration

What are the order of the error for the Trapezoidal and Simpson's method of numerical integration? What is the definition of order of the error of a quadrature formula? Is it true that order of ...
3
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1answer
28 views

Approximation formula on a surface

[Beginning calculus question.] We can get an approximation to the value of a function of two variables, I think, by saying $$ f(a+\Delta x , b+ \Delta y) \approx f(a,b) + f_x(a,b)\Delta x ...
2
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1answer
37 views

Approximating logarithmic series

I'm trying to solve the below recurrence $$T(n) = 2 T(n/2) + n/\log n$$ I referenced some online articles they all are approximating $n \sum_{i=0}^{h-1} \frac{1}{\log n - i}$ to $n ...
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2answers
70 views

Approximating a Harmonic Sum

The infinite sum $\sum_{n=1}^{\infty}\frac{1}{n}$ diverges. However, it is possible to find bounds from some $n$ to another integer $n$. Wolfram alpha is able to give a decimal approximation of the ...
2
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2answers
170 views

Approximation of $\pi$ using Brahmagupta's Identity

Brahmagupta, an ancient Indian Mathematician, gave an pretty efficient algorithm for finding integer solutions to the famous Pell's Equation, far before Fermat propounded this before the European ...
2
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1answer
34 views

Finding $w_1,w_2,b$ such that $w_2 \sigma(w_1 x + b) \approx x$

Let $\sigma(z) = 1/(1+e^{-z})$. How can I find $w_1, w_2, b$ such that $w_2 \sigma(w_1 x + b) \approx x$ for $x \in [0,1]$? The hint provided was to rewrite $x = \frac{1}{2}+\Delta$, assume $w_1$ is ...
1
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1answer
51 views

Probability that sum of two random variables is $1$

Let $X$ and $Y$ be two independent random variables with density functions $f_x(x) = x\exp\left(-\frac{1}{2}x^2\right) \mathbb{}$ where $x \in \mathbb{R}^+$ and $f_y(y) = \frac{1}{2}$ with $y \in ...
2
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0answers
28 views

Meaure-theoretic induction: Why dyadic approximation?

In measure-theoretic induction proofs we always use the dyadic approximation of a non-negative measurable function $Y$ as $$Y_n = \sum_{k=0}^{n2^n-1} k/2^n 1\left(\frac{k}{2^n} \leq Y < ...
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1answer
40 views

How are polynomials graphs approximated?

Say I have the data: $x=[ 1, 2, 3.3, 4, 5.5, 8, 9, 10.2, 11, 45 ]$ $y=[ 9,27,64,91,164,330,462,540,630,10218]$ The data is subjective though. How would one approximate a valid polynomial for this ...
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4answers
637 views

How can I calculate or at least approximate the sum?

As a part of a complexity analysis of the algorithm, I have to calculate the following sum: $$n^{1/2} + n^{3/4} + n^{7/8} + ...$$ where in total I have $k$ elements to sum: ...
0
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1answer
20 views

error bound for polynomial interpolation with derivative matching

We all know the following formula for the maximum error (evenly spaced) polynomial interpolation: $|f(x) - p_n(x)| \leq \frac{h^{n+1}}{4(n+1)} \max_{x\in [a,b]} f^{(n+1)}(x)$ where $p_n(x)$ is the ...
2
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1answer
46 views

Ways to generate triangle wave function.

I recently when searching for parameters on a unit cube in $\mathbb{R}^9$ (we all have our more or less peculiar hobbies, don't we?) found a practical reason to implement a triangle wave function ...
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0answers
28 views

Normalization in least-p'th minimax algorithm

In the book "Practical Optimization: Algorithms and Engineering Applications", the least-$p$th minimax algorithm is presented, for approximation of the minimax optimizer (Alg. 8.1): $Loss_x(k)$ = ...
2
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3answers
66 views

Bounds on Gaussian infinite sum

What are some good upper and lower bounds on the following sum? $$S=\sum_{n=-\infty}^{+\infty}\dfrac{1}{\sigma \sqrt{2 \pi}}e^{-\frac{1}{2}\left(\frac{n}{\sigma}\right)^2}$$ I am looking for ...
7
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3answers
463 views

After 6n roll of dice, what is the probability each face was rolled exactly n times?

This is closely related to the question "If you toss an even number of coins, what is the probability of 50% head and 50% tail?", but for dice with 6 possible results instead of coins (with 2 possible ...
2
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1answer
62 views

Exponential series approximation

I have a series of the following form: \begin{align} \sum_{k=2}^\infty \left( 1 - e^{-ns^{k-1}} \right)^k \end{align} where $0<s<1$. I would like to compute an approximation of this series, for ...
2
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3answers
56 views

Approximation by using Taylor Polynomials - why?

Could anyone tell me why would I want to approximate a function $f$ by using its Taylor expansion (is it the same as saying approximation by Taylor polynomials?), if I have the exact formula of the ...
1
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1answer
42 views

(Ab)using the factorial and gamma functions

I have a product of the following form: $$ P = (N-\alpha)(N-2\alpha)\cdots(N-k\alpha) $$ where $k$ is an integer between $1$ and $N-1$, and $\alpha$ is a real number in $(0,1]$. Clearly, for ...
0
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1answer
23 views

Minimizing sums of values versus minimizing cubes of sums.

I am attempting to find the best path from start to finish from a set of points. Say that one path has costs $x_1,x_2,...,x_n$ and $y_1,y_2,...,y_n$ associated with it. I am attempting to find the ...
1
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2answers
37 views

Complex Equation Formula

Can someone show me how the following two expressions are equivalent: $$\Gamma = \frac{i X - R_c}{i X + R_c} = -e^{-i 2 \mathrm{tan}^{-1} (\frac{X}{R_c})}$$ I'm working through a calculation and I ...
2
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0answers
27 views

Approximation of integrable function by polynomials

Assume $f\in \mathscr{R}(\alpha)$ on $[a,b]$, and prove that there are polynomials $P_n$ such that $$\lim \limits_{n\to \infty}\int_{a}^{b}|f-P_n|^2d\alpha=0.$$ Proof: Let $\varepsilon>0$ be given ...
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2answers
33 views

Approximate a summation

Approximate $3+ \displaystyle \sum_{x = 2}^{999}\dfrac{3(1000-x)}{1000+x}$. It may help to know that $\ln 2 = 0.69$. I was thinking of doing the integral test to approximate this but I am unsure if ...
2
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3answers
55 views

Why does $p$ have to be moderate in the Poisson approximation to binomial random variable?

So the proof that a binomial rv with large $n$ approximates a poisson rv with $\lambda = np$ (given below) doesn't seem to use the fact that $p$ is moderate/small, so why does wikipedia and my ...
3
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3answers
61 views

Numerical method for approximating the standard Normal distribution cdf with mean 0 and variance 1

The standard Normal distribution probability density function is $$p(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2},\int_{-\infty}^{\infty}p(t)\,dt = 1$$ i.e., mean 0 and variance 1. The cumulative distribution ...
0
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1answer
34 views

Interpolation and divergence of $n$-th derivative

i have a curiosity. Let's say i have a function $f \in C^{\infty}[a,b]$ such that there's a $x_0 in [a,b]$ that makes and $f(x_0) = 0$ and $a_n = f^{(n)}(x_0)$ diverges (i.e. $|a_n|=\infty$) could ...
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1answer
31 views

Solution to $0= c_1 x+c_2 x \ln \left(\frac{1+x}{x} \right)+c_3 \ln\left(\frac{1+x}{x} \right)+c_4$

We want to solve \begin{align} 0= c_1 x+c_2 x \ln \left(\frac{1+x}{x} \right)+c_3 \ln\left(\frac{1+x}{x} \right)+c_4 \end{align} for $x \in [0,1]$ and where $c_1,c_2,c_3,c_4$ are non-zero constants. ...
10
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2answers
275 views

Proof of $\sum_{n=1}^{\infty} \frac{x^n \log(n!)}{n!} \sim x \log(x) e^x$ as $x \to \infty$

Prove that $$\sum_{n=1}^{\infty} \frac{x^n \log(n!)}{n!} \sim x \log(x) e^x \,\,\,\text{as}\,\,\, x \to \infty$$ and $$\sum_{n=1}^{\infty} \frac{(-x)^n \log(n!)}{n!} \to 0 \,\,\,\text{as}\,\,\, x \to ...
2
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1answer
158 views

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 ...
2
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1answer
44 views

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 ...
2
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0answers
22 views

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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0answers
34 views

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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0answers
28 views

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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1answer
24 views

Finding the error in a two-step finite difference numerical approximation

I got the following question in a math lecture the other day, and I'm not really sure how to go about it: A differential equation is given in the form $$\frac{\partial y}{\partial x} = f (x, ...
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1answer
32 views

Showing this approximation holds

We see from the formula $(1+t)^2=1+2t+t^2$ that for small $t$, we have the approximate equality $$(1+t)^2\approx 1+2t$$ hence for small $u$, we have $$\sqrt{1+u} \approx 1+\frac u2$$ I know that ...
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0answers
4 views

Approximating semicontinuous functions by continuous functions. [duplicate]

Let $f=f(x):[0,1]\to\mathbb{R}$ be a upper (or lower) semicontinuous function, i.e., $$\limsup_{j\to\infty}f(x_{j})\le f(x)\quad\text{for $x_{j}\stackrel{j\to\infty}{\longrightarrow}x$}$$ (or ...
0
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1answer
36 views

Find the formula $\int_0^1f(x)dx\approx A_0f(0)+A_1f(1)$ that is exact for all functions of the form $f(x)=ae^x+b\cos(\pi x/2)$.

Find the formula $\int_0^1f(x)dx\approx A_0f(0)+A_1f(1)$ that is exact for all functions of the form $f(x)=ae^x+b\cos(\pi x/2)$. I'm not sure how to go about this. Any solutions/hints are greatly ...
1
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1answer
55 views

Verify the polynomial of degree $≤ 4$ is exact: $\int_0^1f(x)dx ≈ {1\over90}[7f(0) + 32f({1\over 4}) + 12f({1\over 2}) + 32f({3\over 4}) + 7f(1)]$.

Verify that the following formula is exact for polynomial of degree $≤ 4$: $\int_0^1f(x)dx ≈ {1\over90}[7f(0) + 32f({1\over 4}) + 12f({1\over 2}) + 32f({3\over 4}) + 7f(1)]$. I'm not sure how to go ...
2
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2answers
33 views

upper bound $\sum_{i=1}^n a_i (a_i-1)/2$ using a function of $\sum_{i=1}^n a_i$

Given $a_i \in \mathbb{N}\cup\{0\}$ and define $$ A(n) = \sum_{i=1}^n a_i (a_i-1)/2 $$ and $$ B(n) = \sum_{i=1}^n a_i $$ Any ideas how to upper bound $A(n)$ as a "function" of $B(n)$? (the tighter, ...
2
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1answer
65 views

Approximating the compond interest for a loan

A young boy (13 years old), son of friends of mine, is already very dedicated to mathemetics. He told me that, in the classical formula $$A=P\frac{i \,(i+1)^n}{(i+1)^n-1}$$ using his calculator he was ...
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0answers
19 views

Successive Approximation Algorithm for Optimal Stochastic Control: toy example problem

In https://drive.google.com/file/d/0B5kp8BrW_9rdZTBERzNmQnRKQjA/view?usp=sharing (A successive approximation algorithm for stochastic optimal control) by Chang and Krishna an algorithm is described ...
1
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1answer
28 views

Show $\int_0^tK(s) e^{-k(t-s)} \text{d}s\approx K(t)\int_{-\infty}^t e^{-k(t-s)} \text{d}s$ using Watson's lemma

How can you show using Watson's lemma, that for some infinitely differentiable function $K(s)$ and $ kt \gg 1$ that $$\int_0^tK(s) e^{-k(t-s)} \text{d}s\approx K(t)\int_{-\infty}^t e^{-k(t-s)} ...
0
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1answer
48 views

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 ...
0
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0answers
8 views

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 ...
1
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1answer
36 views

How to prove/disprove $ \sum_{i=1}^{n} \frac{a_i}{ \sum_{j=1}^{i} a_j } \approx \log \sum_{i=1}^{n} a_i, \quad a_i \in \mathbb{N}^+ $?

Remember $\sum_{i=1}^{n} 1/i$ is asymptotic to $\log n$. Is it possible to generalize it to the following?: $$ \sum_{i=1}^{n} \frac{a_i}{ \sum_{j=1}^{i} a_j } \approx \log \sum_{i=1}^{n} a_i, \quad ...