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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27 views

Using the Maclaurin series to approximate $f(0.1)$ for $f(x)=(3+e^{2x})^{0.5}$

I was tasked to use the Maclaurin series to calculate $f(0.1)$ of $f(x)=(3+e^{2x})^{0.5}$. I got the Maclaurin expansion of $p_2(x) = \sqrt{3} + 4x +5x^2$ into which I plugged $0.1$ to yield ...
4
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0answers
102 views
+50

Differences among Cauchy, Lagrange, and Schlömilch remainder in Taylor's formula: why is generalization useful?

I would like to know what really are the main differences (in terms of "usefulness") among Cauchy, Lagrange, and Schlömilch's forms of the remainder in Taylor's formula. Could you provide examples ...
0
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0answers
140 views

Upper bound for L1-L2 optimization problem

I am interested in the following convex optimization problem: \begin{align*} \max & \lVert x \rVert_1 \\ \text{s.t.} & \lVert x -a\rVert_1 \le K \\ & \lVert b\circ x\rVert_2 \le 1\\ & ...
1
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2answers
29 views

Spline approximation for $g(t) = \frac{t e^{-t}}{(x+t^2)^2}$

Is there any nice way to do a spline approximation for $$ g(t) = \frac{t e^{-t}}{(x+t^2)^2}\,, $$ where $x$ is some constant? I tried finding nice interpolation points, however this proved very ...
1
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1answer
16 views

Locally compact metric space, Urysohn, approximation

Let $E$ be a locally compact separable metric space, $\mathcal{B}(E)$ be the $\sigma$-algebra of $E$ and $m$ be a $\sigma$-finite borel measure on $(E,\mathcal{B}(E))$. Assumtion There exists a ...
0
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0answers
11 views

Flipping X, Y Known Values with Result Values; Table Data and Linear Interpolation

I am not knowledgeable in the terminology I need to be searching for to accomplish what I need in Excel. I have the following table of values which gives me the resulting RPM if I know the Pressure ...
16
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3answers
334 views

Approximating $100!$

I participated in an Estimathon (a speed contest of Fermi problems) not long ago. It works as follows. Contestants are given questions and they must give a closed range $[a,b]$ which should contain ...
0
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1answer
54 views

If a continuous function on $[0,\pi]$ integrates to zero against cosines, it is identically constant

Let $f(x)$ is Continuous function on $[0,\pi]$,and for $n=1,2,.....,$ the function $f(x)$ has the following property:$$\int_{0}^{\pi}f(x)\cos{(nx)}dx=0.(n=1,2,......)$$ Proof: $f(x)\equiv C$(C is ...
5
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2answers
86 views

Prove that $\int_0^\infty\frac{x^n}{1+e^{x-t}}\mathrm{d}x = \frac{t^{n+1}}{n+1} + o(t^n)$, when $t \to \infty,\,n\in\Bbb{R}^+$

I hae to prove that $$\int_0^\infty\frac{x^n}{1+e^{x-t}}\mathrm{d}x = \frac{t^{n+1}}{n+1} + o(t^n), \quad\text{ when } t \to \infty,\,n\in\Bbb{R}^+$$ where $o(\cdot)$ is the Little-o notation. What ...
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0answers
7 views

Probability distribution of the number of Dirichlet’s Approximations of random reals as a function of $M$

I curious about the Probability distribution of the number of Dirichlet’s Approximations of random reals as a function of $M$. What I mean by this is the following: We fix some integer $M$, and we ...
1
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1answer
36 views

Approximation by polynomials

I know the Approximation Theorem of Weierstrass. I think one can apply it to my question but I don't see directly how. Assume $f$ is a continuous function on the unit interval $[0,1]$ such that ...
2
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0answers
75 views
+50

Taylor Formula: Lagrange's remainder vs Cauchy's remainder (and other less known forms)

While solving problems and exercises, so far I've only used Lagrange's form of the remainder. Indeed, it must be said that many textbooks don't even mention other forms of the remainder for Taylor's ...
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0answers
24 views

the approxmiation polynomial in infinite norm.

$\mathbb{P}_n$ is the set of polynomials with order at most $n$ on $[a,b]$. $f$ is a real function on $[a,b]$. Suppose there exist $n+2$ partition points $a\leq x_1\leq x_2\leq\cdots\leq x_{n+2}\leq b ...
1
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1answer
54 views

Proofs involving positive real numbers

I have two questions related to positive real numbers: If a and b are two vectors of positive random integers (no specific statistical distribution) and size N by 1 , we want to prove that the inner ...
1
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2answers
465 views

How to find upper bound on absolute error with composite trapezoid rule

Obtain an upper bound on the absolute error when we compute $\displaystyle\int_0^6 \sin x^2 \,\mathrm dx$ by means of the composite trapezoid rule using 101 equally spaced points. The formula I'm ...
1
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1answer
14 views

Taylor expansion of a logaritmic function

A function is given as $ln (y) = ln(\alpha)-\frac{\lambda}{\gamma}ln(\delta L^{-\gamma}+(1-\delta)K^{-\gamma})$ I need to find the second order Taylor $ln(y)$ around $\gamma=0$. How can it be done ...
1
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1answer
19 views

Normal approximation of Poisson Distribution

Hi currently studying for a final exam and I just want to confirm my approach/answers to this problem are correct: Suppose that $X \sim \mathrm{Poisson}$. We wish to test $H_0: \lambda = 50$ vs ...
1
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2answers
27 views

Polynomials of best approximation

The question is about approximating the continuous function in an interval $[a, b]$. If we consider the linear space of all such functions endowed with the norm $$||f|| = \max_{x \in [a, b]}|f(x)|$$ ...
3
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1answer
67 views

How can I recursively approximate a moving average and standard deviation?

Consider a sequence of measurements $(x_1, x_2, ...)$. Let $\mu_n$ be the $p$-period moving average defined by $$\mu_n = \frac{1}{p}\sum_{i=n-p+1}^nx_i$$ and $\sigma_n$ be the $p$-period moving ...
0
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1answer
570 views

How to calculate APR using Newton Raphson

I'm have a computer program to calculate apr using Newton Rhapson. I imagine most mathletes can code so i dont imagine the coding being an issue. The solution is based on this initial formula ...
1
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2answers
29 views

Finding the minimum point looks easy with a graph but hard with a formula

My research has lead me to the following function: $$ \frac{\sin(x) [\sin^2(x)\cdot F+ \cos^2(x)/F ]} { 1 - \cos(x) } $$ $F$ is a parameter, and I would like to find the minimum value of this ...
0
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2answers
43 views

What are some quickly convergent, easily calculated approximations for common functions for when you've forgotten a calculator to a test?

I think it's nice not to rely too much on a calculator, whether it's forgotten or forbidden. Approximations can be useful on exams when you want a good guess at the answer to see if it's somewhat ...
0
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1answer
24 views

Time series closed form similar to harmonic series

I want to find the closed form for the following: $$ S(k, \alpha) = \sum_{t=T-k}^T \frac{\alpha^{-t}}{t} $$ when $\alpha \in (0,1)$. For harmonic series there is an easy way to upperbound: $$ H(k) ...
0
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2answers
36 views

Converting 29^1312000 to base 10

I am trying to do some calculations with the number 291312000 and I find it would be much easier if I could convert it (approximately) to a base 10 number. The closest I could come was to start with ...
12
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4answers
274 views

Which expansion of $e$ is more accurate?

We have two forms of $e^x$ $$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+....$$ and $$e^x=\frac{1}{\displaystyle 1-x+\frac{x^2}{2!}-\frac{x^3}{3!}+....}$$ The second form comes from ...
0
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0answers
18 views

Polar coordinate for complicated curves

In general polar representation of a closed curve is done by coordinate $(\theta,r(\theta))$, $\theta\in (0,360)$. When working with real data, I got a closed curves whose graph looks like the below ...
50
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4answers
4k views

Motivation for Ramanujan's mysterious $\pi$ formula

The following formula for $\pi$ was discovered by Ramanujan: $$\frac1{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}\!$$ Does anyone know how it works, or ...
2
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2answers
473 views

Approximate a polynomial function using a sum of sine waves

I have a polynomial function which I need to approximate by a sum of sine waves with constant amplitude along a given domain. From what I hear, this might be a good time to make use of Fourier ...
0
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1answer
17 views

Series and quadratic approximation

Been working out series for the past day and have come across this question. I would kindly ask how I ought to tackle it, I've done Binomial and Maclauren's however this seems to require another ...
1
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0answers
31 views

Closed forms for two times series similar to geometric series, but with additional power

Does anyone know a close form solutions to any of the following time series? (approximate upper bounds might as well work). $$ \sum_{k=1}^T \frac{1}{2^{k^2}} $$ or $$ \sum_{k=1}^T k ...
2
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0answers
28 views

Continued fraction approximation to a function and its derivative

I am recently working on fitting a model with incomplete beta function. In order to put it into my optimization algorithm, I must find out the derivatives of the incomplete beta function $B_p(x,y)$ ...
2
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1answer
44 views

Applying Newton-Raphson method to $a\cdot b^{-2}=c\cdot d^4+e\cdot f(d)$

I am familiar with the method and it's application in classic problems, but I have troubles tackling the function I need to solve with it. So, variables in problem: Real numbers, all are known ...
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0answers
12 views

Approximating $\int_0^1 t^n \exp \left( -\frac{(t-1/2)^2}{2\sigma^2}\right) \, dt$ without Laplace's method

I'm doing some finite element analysis, and one frequent integral that I need to evaluate is, $$\int_0^1 t^n \exp \left( -\frac{(t-1/2)^2}{2\sigma^2}\right) \, dt$$ for $\sigma \in \mathbb{R}$ and ...
0
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0answers
27 views

Additive and Multiplicative Error in $n!$ Approximation

Let $S(n)=\sqrt{2\pi n}\big(\frac{n}{e}\big)^n$ be the approximation of interest to $n!$. What are good lower and upper bounds on the following two functions $$(1)\mbox{ }|S(n)-n!|?$$ $$(2)\mbox{ ...
2
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2answers
235 views

Approximate continous function with linear growth condition by Lipschitz function

Suppose a continuous function $f(u)<K(1+|u|)$ for some positive number $K$. How can we find a sequence of Lipschitz functions $f_{n}$ that converge to $f$ uniformly on $\mathbb{R}$. If we require ...
1
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1answer
38 views

Derivative of $\|Ax-b\|_1$

Using least squares approximation $E^2 = \| Ax - b\|^2 = (a_1x - b_1)^2+...+(a_mx-b_m)^2$ The derivative of E^2 at the point $\hat{x}$ is zero if: $(a_1\hat{x}-b_1)a_1+...+(a_m\hat{x}-b_m)a_m=0$ ...
0
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0answers
18 views

Laplace's method with nontrivial parameter dependency

I need to approximate the following integral using Laplace's method: $$ \int_0^{\infty} \frac{x^{\lambda} \lambda^{-x}}{(1+x^2)^\lambda} \\ = \int_0^{\infty} \exp\left(\lambda \log(x) - ...
1
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1answer
32 views

Physics Approximations Quadratic Equation

I'm having a hard time following one of the solutions to this physics problem. In particular, the math. Consider, $$a\Omega ^2 + b\Omega + c = 0$$ The solutions to this quadratic equation are, ...
1
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1answer
42 views

Least absolute deviation for item prices

How would I calculate the values of $A,B,C$ using least absolute deviation? $R = 1$ $2A + B$ = $C + R$. $B + C$ = $5A$. $A + C + 2R$ = $B + 4R$. $A + B + C$ = $6.33R$. Using least squares ...
1
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1answer
24 views

Stirling formula on $\frac{\gamma(\frac{1}{2}s)}{\gamma(\frac{1-s}{2})} = O(|t|^{\sigma - \frac{1}{2}})$

I tried to prove $\frac{\Gamma(\frac{1}{2}s)}{\Gamma(\frac{1-s}{2})} = O(|t|^{\sigma - \frac{1}{2}})$ using stirling formula when $s =\sigma+it$. However, since stirling formular for gamma function is ...
2
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4answers
49 views

How to extract fraction from a floating point number

I'm making some tests with float type (floating point number) with programming and in some of my tests I need to extract the fraction that originates the float value. Let $ x $ be a floating point ...
3
votes
1answer
82 views

If $f \in\operatorname{Lip}_K[a, b]$, show that $f$ can be uniformly approximated by polynomials in $\operatorname{Lip}_K[ a, b]$.

If $f \in\operatorname{Lip}_K[a, b]$, show that $f$ can be uniformly approximated by polynomials in $\operatorname{Lip}_K[ a, b]$. Context: $f \in \operatorname{Lip}_K[a,b]$ then it is ...
1
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2answers
37 views

Least Squares approximation for item prices

Let's say that $A$, $B$, $C$ are different items with different values. $R$ is a unit of currency, for simplicity I'll let it be $1$. Traders frequently trade these items on an open market. Price is ...
16
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3answers
1k views

Sine Approximation of Bhaskara

An Indian mathematician, Bhaskara I, gave the following amazing approximation of the sine (I checked the graph and some values, and the approximation is truly impressive.) $$\sin x \approx ...
0
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3answers
40 views

Notation for asymptotic approximation

I was reading Stirling's approximation and got quite confused with the idea of asymptotic formula. So in Wikipedia it says that a function $F(n)$ of $n$ is asymptotic formula for $P(n)$ if $P(n)$ is ...
0
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1answer
29 views

Approximating a binomial coefficient using Stirling's formula

I am working on a problem of modelling a rubber molecule as a one-dimensional chain consisting of $N=N_{+}+N_{-}$ links, where $N_{+}$ points in the positive $x$-direction a distance $a$ and $N_{-}$ ...
1
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0answers
26 views

$L^2$ inequality for derivatives of polynomials on triangles

I'm reading a paper which states the following inequality, but the (presumably) elementary proof is cited to be in a document, which is too old to get access to. Let $p: \mathbb{R}^2 \to \mathbb{R}$ ...
1
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2answers
105 views

How come $\pi$ is usually approximated as 3.14 or 22/7?

I've heard that $\pi$ is usually approximated as 3.14, but it can also be approximated as 22/7, which is equal to 3.142857142857142857.... Guess what? $\pi$ can also be approximated as 355/113, ...
0
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0answers
15 views

Approximation for uniform load on parabolic cable along its arc length

I am doing analysis for cable structures. Let's say that the cable stretches from point A to point B and carries a vertical ...