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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2
votes
0answers
33 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 < ...
1
vote
1answer
45 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 ...
5
votes
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
votes
1answer
28 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
votes
1answer
52 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 ...
1
vote
0answers
29 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
votes
3answers
77 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
votes
3answers
483 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
votes
1answer
66 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
votes
3answers
65 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
vote
1answer
51 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
votes
1answer
25 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
vote
2answers
40 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
votes
0answers
29 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 ...
-1
votes
2answers
34 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
votes
3answers
60 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
votes
3answers
81 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
votes
1answer
38 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 ...
-1
votes
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
votes
2answers
279 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
votes
1answer
160 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
votes
1answer
52 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
votes
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 ...
1
vote
0answers
36 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 ...
0
votes
0answers
38 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 ...
0
votes
1answer
31 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, ...
0
votes
1answer
34 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 ...
0
votes
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
votes
1answer
45 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 $f(x)=ae^x+b\cos(\pi x/2)$. [closed]

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 ...
0
votes
1answer
63 views

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

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
votes
2answers
34 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, ...
3
votes
1answer
116 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 ...
0
votes
0answers
23 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
vote
1answer
30 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
votes
1answer
51 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
votes
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
vote
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 ...
1
vote
1answer
46 views

Can you determine the average second derivative from a set of points?

Let us say we have a smooth function $f$. We can find the exact average of $f'$ on the interval $[a,b]$ via $$\bar{f'}=\frac{f(b)-f(a)}{b-a}$$ My question is, can you find the exact average of the ...
0
votes
0answers
69 views

is it true that $ \frac{1}{\epsilon}\ln (1- \epsilon x) \approx - \frac{1}{1-\epsilon}x $?

Target is to approximate $\frac{1}{\epsilon}\ln (1- \epsilon x) $ ($\epsilon, x \in (0,1) $). Here is one using $\ln (1+y) \approx y $: $$ \frac{1}{\epsilon}\ln (1- \epsilon x) \approx -x $$ I ...
1
vote
1answer
20 views

Avoid loosing precision with the Binomial Distribution /Bayes?

So when dealing with the Bayes' Rule and Binomial distributions, the value $p^k(1-p)^{n-k}$ loses precision and becomes 0 when $n$ and $k$ are large(noting that the binomial coefficient can be safely ...
0
votes
2answers
39 views

How to calculate parameters of a logarithmic approximation trendline?

I have a set of (Y) data $\left\{y_1, y_2, ..., y_n \right\}$ and a set of (X) $\left\{x_1, x_2, ..., x_n \right\}$ which I use to build a graph. I need to place a logarithmic trendline over the ...
3
votes
1answer
74 views

Tight approximation of a Torus Knot length

Is there a simple formula for a tight approximation of the torus knot length ? (specifically a formula that does not involve integrals or any iterative procedures). The torus knot parameters are $(p, ...
0
votes
1answer
76 views

Matlab code, approximate an integral using Monte-Carlo method.

so i have to program the approximation of these two integrals using Monte-Carlo method: $$\int\int_D e^{x^2+y^2} \, dy \, dx $$ $$D=\{(x,y) \in \Bbb R \mid x^2+y^2\le9\}$$ and: $$\int_0^2 ...
3
votes
1answer
54 views

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 ...
0
votes
1answer
98 views

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?
1
vote
2answers
52 views

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 ...
1
vote
0answers
40 views

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) ...
2
votes
3answers
32 views

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, $\sqrt{(1+\Delta^2)} = (1+\frac12\Delta^2)$ ...
2
votes
1answer
45 views

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 ...
1
vote
1answer
49 views

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.