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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10
votes
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
votes
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
votes
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
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
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 ...
0
votes
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 ...
0
votes
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, ...
0
votes
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 ...
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
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
vote
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
votes
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
votes
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 ...
0
votes
0answers
20 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
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
votes
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
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
28 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
48 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
19 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
28 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
49 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
58 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
0answers
49 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
92 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
34 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
27 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
42 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
44 views

Let $f(x)=1/x$ and prove that $f[x_0,x_1,…,x_n]=\prod_{i=0}^nx_i^{-1}$.

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.
1
vote
1answer
31 views

How to find the closest bounded rational approximation to a rational number?

Say I have a rational number $a/b$ and I want to find its closest rational approximation $x/y$ where $$x_- \leq x \leq x_+$$ $$y_- \leq y \leq y_+$$ for some constants $x_\pm$, $y_\pm$. How can I ...
0
votes
0answers
18 views

Ortogonal polynomial regression

I need to fit a weighted set of points with orthogonal polynomials based on least squares polynomials. The grid on X is nonuniform, so Chebyshev polynomials can't be applied. The data points are like ...
1
vote
1answer
29 views

Computing Two-Norm for interpolation of functions

$$ f(x) = x^3, p(x) = (3/2)x^2 − (1/2)x $$ The two-norm of f(x) - p(x) is: $$( \int_0^1 (f(x) - p(x))^2 dx )^{1/2} $$ p(x) interpolates f(x) at $$x=0, x=1/2, x=1$$ The result of the two-norm ...
0
votes
2answers
37 views

Notation and semantic for approximation?

Likewise the notation $\approx$ that says is "approximately equal to.." is there a symbol for the meaning "is approximately less of..."? What's the formal meaning of such symbols?
1
vote
1answer
15 views

Two dimensional taylor expansion of arbitrary function

Consider the function dependent on the variables $N_t$ and $N_{t-1}$. Call the function $f$ so $f = f(N_t, N_{t-1})$. Now suppose we could write $N_t = N^*+n_t$ where $N^*$ is constant, and $n_t$ ...
1
vote
2answers
42 views

Approximate an exponential factor

What math methods can I use to approximate lambda in the following system of equations?: $$ e^{-0.05\lambda}=0.5469\\ e^{-0.1 \lambda} = 0.3229\\ ...\\ e^{-0.2 \lambda} = 0.1226$$ I am trying to fit ...
1
vote
2answers
78 views

Finding root of $\cos(x)$ by Newton-Raphson method

The exercise asks me that if I want to find the root of $f(x) = \cos(x) = 0$ using Newton-Raphson method, does the initial value matters? I know that Newton-Raphson method is a special case of the ...
0
votes
1answer
32 views

Real Analysis: Bounds for derivatives using Taylor's Theorem

Suppose that $f''$ exists on [0,1] and that $f(0)=0=f(1)$. Suppose also that $|f''(x)|\leq K$ for $x\in(0,1)$. Prove that $|f'(1/2)|\leq K/4$ and that $|f'(x)|\leq k/2$ for $x\in(0,1)$. I'm trying to ...
1
vote
1answer
39 views

Approximation of power with binomial theorem [closed]

How to approximate value of $0.99^9$ with binomial theorem? (with accuracy $10^{-3}$)
2
votes
0answers
35 views

Textbook on Approximation

Does anyone know of a good book on approximations methods? In particular finding approximations to sums, integrals, etc via upper/ lower bounds, power series approximations, etc and bounding the ...
1
vote
1answer
30 views

An approximation for the Lambert W-function

Proposition Let $f(x) = k^{x}x$, where the values of both $f(x)$ and $k$ are known. Let $x_{0} = f(x)$, and: $$x_{n + 1} = \frac{1}{2}\log_{k}{\left(\frac{k^{x_{n}}x_{0}}{x_{n}}\right)}$$ ...
0
votes
1answer
41 views

What is my mistake: Asymptotic behaviour of the following integral?

Okay, I am going to be very specific. I have the following integral $$\int_{-1}^1 \mathop{dx}\frac{x^{n-2m}(a^2+x^2)^{(k-2)n/2+(3-k)m/2}}{(c_1 ...
2
votes
0answers
24 views

Approximation of vector-valued Lipschitz functions

I am looking for methods of approximating vector-valued Lipschitz functions up to arbitrary precision, by convolution with heat kernels in particular. Here is what I am interested in. Let $f$ be a ...
0
votes
0answers
42 views

How to prove that problem is NP-hard by making a reduction?

Convering by triples Data: A set Y of cardinality 3n and a family C = ($C_{1},...C_{m}$) of triples of elements of Y: for all i, $C_{i}$ $\subset Y$ and |$C_{i}$| = 3. We admit that COVERING BY ...
1
vote
1answer
84 views

Estimate the mode of the binomial distribution without Stirling's formula

Context: Let $\tilde B_n$ be standardized binomial distributed with $p\in(0,1)$ be the probability of success in the $n$ Binomial trials. So $P(\tilde B_n = x_k)=\binom nk p^k q^{n-k}$ for $x_k = ...
0
votes
0answers
27 views

Approximation algorithm for ratio is $2$?

Let consider a graph $G$ and a subset of its vertices. Let $G_s$ be the complete graph on $S$ with the following weights on its edges: the weight of $(x,y)$ is equal to the length of the shortest path ...
0
votes
1answer
21 views

What are the initial and boundary conditions for this problem?

I'm trying to solve a heat diffusion problem (with finite difference approximation) with conditions stated like this: A 10cm thin plate is initially at $120^{\circ}C$ then suddenly the right ...
1
vote
1answer
23 views

Error Propagation

I hope I am right in this section. I am unsure with error propagation. When calculation the error in a titration, many errors has to be taken into account: Error in Glassware/ Error in Balance/ ...
2
votes
1answer
42 views

check that $\int_0^{\pi/2}\frac{1}{\sqrt{1-k^2sin^2\theta}}d\theta$ is equivalent to $-\log\sqrt{1-k^2}$ for $k\to1$. [duplicate]

Please help check that $\int_0^{\pi/2}\frac{1}{\sqrt{1-k^2sin^2\theta}}d\theta$ is equivalent to $-\log\sqrt{1-k^2}$ for $k\to1$. I really have no clue how to do this. Could anyone kindly help or ...