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
43 views

Using $\ln (\cos x)=\frac{-x^2}{2}-\frac{x^4}{12}+…$, approximate $\ln 2$ in terms of $\pi$

Using $f(x)=\ln (\cos x)=\dfrac{-x^2}{2}-\dfrac{x^4}{12}+\dots $, approximate $\ln 2$ in terms of $\pi$. I know $\cos(x)$ will never be two - so what can I actually substitute in to get something ...
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0answers
26 views

Bernstein approximation on the simplex [on hold]

As we all know, for some univariate monomial $x^{m}$ defined on the [0,1], we can get its Bernstein approximation of order $d$, which is ...
7
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0answers
122 views
+100

Approximation of the exponential

Let $c>1,k\in\mathbb{N}$. Let's consider two approximations of the exponential function : The first one is the most common one $f_k(x)=\left(1+\frac{x}{c^k}\right)^{c^k}$ and the second one is ...
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0answers
9 views

Ratio of fpras approximations

If I need to compute the ratio $\frac{A}{B}$ and if there exists an FPRAS that approximates the numerator and the denominator separately, that is, $\exists A_{fpras},B_{fpras}$: $Pr(A(1-\epsilon)\le ...
1
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1answer
16 views

polynomial approximation - basic chebyshev question

I was asked to find the best linear approximation to $f(x)=x^2$ in $x \in [0,1]$ using chebyshev polynomials, meaning, using the known property that $2^{1-n}T_n(x)$ is the best approximation to $0$ at ...
3
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0answers
29 views

Function Looks Poisson-Like: But What's the Parameter $\lambda$?

(On pause) I have $$f\left(x\right)=-x\left( x\sqrt{4-x^2}-4\arccos\left(\frac{x}{2}\right) \right)\arccos\left(\frac{x^2+d^2-1}{2dx}\right)$$ which looks a bit like the continuous version of ...
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4answers
129 views

What does Pi equal to [duplicate]

What is the approximation of pi in a fraction form. I am very curious to know what it is. I have been seeing pi almost everywhere.
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0answers
22 views

Is there a standard way to obtain an approximation piecewise-linear function for a function

I am trying to find a generic way to get an approximation function for a given function. (I will be doing it programmatically eventually). What I want to obtain is a set of pairs, mapping the x-axis ...
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5answers
62 views

Approximate $\coth(x)$ around $x = 0$

I'm trying to approximate $\coth(x)$ around $x = 0$, up to say, third order in $x$. Now obviously a simple taylor expansion doesn't work, as it diverges around $x = 0$. I'm not quite sure how to ...
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0answers
22 views

Chebyshev polynomials approximation - Is there a way to generalize this

In an exam I was given this question: let $f(x)=x^3$. We want to find the best linear approximation (best in the sense that the maximal error is minimized) of $f$ in the interval $[-1,1]$ using ...
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0answers
41 views

Is the Taylor Expansion a good approximation

Say I use a computer to sum the first 26 terms of $e^{-5}$ (degree 25), will this taylor expansion provide a good approximation? It summed to $.0067$ To me this seems like a good approximation, but I ...
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0answers
12 views

Linear interpolation vs polynomial interpolation

Why linear interpolation is better than polynomial interpolation when we want to approximate $f(0.25)=e^{0.25}$? I can't formulate a concrete explanation. I thought that maybe it has a link with the ...
0
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1answer
36 views

Approximating $a = bx + cx^2 + O(x^3)$

I'm attempting to work out a problem of the form $a = bx + cx^2 + O(x^3)$, where I need to solve for $x$. To be honest, I don't really know how to work this out. Someone suggested that it can be done ...
2
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1answer
27 views

Lower Bound for Bernstein Approximation

If have to do solve the following problem: Let $f(t) = | t - \frac{1}{2} |$ be defined on $[0,1]$ and let $B_n ( t )$ denote the $n$-th Bernsteinpolynomial for $f$, i.e. $$ B_n ( t ) = \sum_{i = 0}^n ...
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0answers
6 views

Approximte curve by lorenzians

I am looking for some input on how to approximate some given (well behaved) function $f(x)$ by a sum of lorenzians: $$f(x)=\sum_{n=0}^{\infty}\frac{a_{0,n}}{(x-a_{1,n})^2+a_{2,n}^2}$$ If possible, I ...
0
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0answers
12 views

Gaussian Q-function

I want a good approximation to 2Q(x)-Q(x)^2. where Q is the Gaussian Q-function. To use it in wireless communications. I found that Q(x)^2 = 1/pi * int(0,pi/4) *exp(-x^2/2sin^(theta))
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0answers
50 views

Is there any way to justify this approximation of the solution of a rather simple equation?

I am considering the problem of finding the root of equation $$f(x)=-x+\sum_{i=m}^{i=n} \sqrt{x+i}=0$$ where $m,n$ can both be from very small to very large integers. Since, in a single calculation, ...
5
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0answers
41 views

Pointwise approximation of a closed operator

If $T:\mathcal D(T) \rightarrow \mathcal Y$ is a closed operator from a Banach space $\mathcal X$ to a Banach space $\mathcal Y$, is it possible to find bounded operators $T_n\in \mathscr B(\mathcal ...
1
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1answer
42 views

Fourier Series and differential equation with epsilon

Happy New Year! I am stuck for days on expressing the solution of a differential equation using Fourier series. The question is: Consider the equation: $$\ddot{x}+x+\epsilon\left(\alpha ...
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3answers
106 views

Evaluate $ \int_0^1 \sqrt{x}\sin(x)dx $ to accuracy 0.001.

Evaluate $$ \int_0^1 \sqrt{x}\sin(x)dx $$ to accuracy 0.001. By definition, there exists an N such that for n > N, $$ \left| \int_0^1 \sqrt{x}\sin(x)dx - \sum\limits_{i=1}^n ...
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1answer
32 views

Machine limit analysis of $\sqrt {x^2-a^2}-(x-a)$

Let $L(x)=\sqrt {x^2-a^2}-(x-a)$. I've been messing around with this equation on the calculator and found out that for certain values of $x$, the equations behave as $x \gg a$. Considering only for $x ...
3
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0answers
50 views

Rational approximation of pi

I found this problem intriguing: $355 / 113 = 3.14159292035398\ldots$ gives the approximation of $\pi$ in $7$ correct numbers, say $C(355/113)=7$, but it number of digits in numerator + number of ...
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0answers
71 views

How to show$\frac{\pi}{4} = \frac{2\cdot4\cdot4\cdot6\cdot6\cdot8 \dotsm}{3\cdot3\cdot5\cdot5\cdot7\cdot7 \dotsm}$?

I am doing the exercises of Structure and Interpretation of Computer Programs. In exercise 1.31 the following equation is casually shown as an approximation of $\pi$: $$\frac{\pi}{4} = ...
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1answer
42 views

Calculating area of complex ploygons (odd shapes)

We need a formula / algorithm to find the area of a shape, based on user selected co-ordinates (mouse click). We know the x and y co-ordinates of each line drawn by the user, then using the pixel ...
0
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0answers
18 views

What conditions are required to approximate an unknown function with trapezoidal rule?

What conditions are required to approximate an unknown function with trapezoidal rule? I have been told that function should be linear, what does that mean? Does it mean for their graph to be a ...
10
votes
2answers
272 views

$e^{\pi\sqrt N}$ is very close to an integer for some smallish $N$s. What about $\pi^{e\sqrt N}$?

Heegner numbers (1, 2, 3, 7, 11, 19, 43, 67, 163 - let's use symbol $H_n$) are know for peculiar property that $e^{\pi\sqrt{H_n}}$ are almost integers: $$e^{\pi \sqrt{19}} \approx 96^3+744-0.22$$ ...
10
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1answer
241 views

How was this approximation of $\pi$ involving $\sqrt{5}$ arrived at?

The Wikipedia article for Approximations of $\pi$ contains this little gem: $$ \pi \approx \frac{63}{25}\times\frac{17 + 15\sqrt{5}}{7 + 15\sqrt{5}} $$ which is clearly in $\mathbb{Q[\sqrt{5}]}$. ...
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0answers
27 views

“Absolute error” limit

I am confused about a section on "Reliable Digits" in a math book, in which the authors says "...the absolute error should not exceed a half unit of the last digit retained." He then gives this ...
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1answer
25 views

X values for least squares approximation

If I have a yearly quantity (eg. 2000 - 45, 2001-67, 2002 - 38.....2010 - 38) and I need to find the least squares line for this relationship, what should I use for the X values? Should I use the ...
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2answers
66 views

How do I see that $1-(1-\frac 1 M)^Q \approx \frac Q M$ (provided that $Q$ is small compared to $M$), where both $Q$ and $M$ are integers?

How do I see that $$1-(1-\frac 1 M)^Q \approx \frac Q M$$ (provided that $Q$ is small compared to $M$), where both $Q$ and $M$ are integers ? The approximation is stated in a book without proof. ...
1
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2answers
61 views

What may be meant by the $\wedge$ here?

I am dealing with article Two Moments suffice for Poisson Approximations: The Chen-Stein Method by Arratia, Goldstein and Gordon. On page 11 there is an expression with a $\wedge$ appearing in it: ...
3
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0answers
104 views

Diophantine approximation with additional constraints

I am trying to compute best rational approximations to various transcendental numbers $c$, subject to the following constraints: $$\frac {i j} {2^k} = c + \epsilon, \space\space2^n \le i, j \lt ...
1
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1answer
16 views

The sequence of functions $u_{n+1}(t)=u_n(t)+\frac{1}{2}(t-u_n^2(t))$ approximates $\sqrt{t}$ from below

The square root function on $[0,1]$ is approximated by a sequence of functions (polynomials) defined on $[0,1]$. The induction hypothesis is that you have functions such that $$0 = u_0\leq u_1\leq ...
24
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4answers
1k views

Prove that $2^{2^{\sqrt3}}>10$

With a computer or calculator, it is easy to show that $$ 2^{2^\sqrt{3}} = 10.000478 \ldots > 10. $$ How can we prove that $2^{2^{\sqrt3}}>10$ without a calculator?
2
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3answers
74 views

Approximate $f''(3)$ from Table of Values of $f(x)$

Considering the table above, what is the best approximation for $f''(3)$? How would I solve?
1
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1answer
60 views

$\dfrac{dy}{dx} + 0.4y = 3e^{-x}$

calculate $y(3)$ using step size $h=1$ given $y(0)=5$ via euler method solve the differential equation calculate the error between the approximation and actual value of $y(3)$ I got, ...
4
votes
2answers
170 views

Is the “almost-identity” $\sum_{k=0}^\infty \left[\pi^{\frac k2}\big/\Gamma{\left(\frac k2+1\right)}\right]\approx46$ significant or a coincidence?

Recently, I read an article about “almost-identities”. It said, that for every “almost-identity” we have to decide whether it is a coincidence or not. By myself, I discovered that $$ \sum_{k=0}^\infty ...
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0answers
41 views

Algorithm-generating algorithm

Is there an algorithm that can create other algorithms based on any number of arguments? For example, a way to determine a function $ f (x) $ from a given input and a given output? I.e. if $ f (2)=4 $ ...
3
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1answer
66 views

Good approximation for $\binom{N}{\frac{N}{2}}$

$$\log_2\binom{N}{\frac{N}{2}}\approx N\log_2N - 2(N-\frac{N}{2})\log_2(N-\frac{N}{2})=N\log_2N - 2\frac{N}{2}\log_2(\frac{N}{2})$$ $$=N\log_2N - {N}{}\log_2({N}) + {N}{}=N$$ ...
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1answer
28 views

Determining a formula to approximate a periodic error

I am working on a barn door tracker for taking astro photos. My drive train has a small periodic error that I'm trying to eliminate and I was hoping someone might be able to suggest a formula that ...
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0answers
27 views

Rewriting the heat diffusion equation with temperature dependent diffusion coefficient to include joule heating.

I am modelling heat flow in a solid round copper conductor with a set area. I plan to discretize and solve numerically in Python. However, I only have a curve fit for thermal conductivity and specific ...
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2answers
31 views

Trying to show property of fractional part

We have this sequence $$\left\{\{\sqrt n\}\right\}_{n=1}^\infty\;,\;\;\{\sqrt n\}:=\;\text{the fractional part of}\;\;\sqrt n$$ The exercise is to prove it doesn't have a limit, and we get several ...
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1answer
35 views

Having trouble combining Weierstrass approximation theorem and the infinite sequence of holomorphic functions

The Weierstrass approximation theorem says that all continuous functions on $[0,1]$ can be approximated uniformly by polynomials. Trying to facilitate the digestion of the fatty Christmas food, I ...
0
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1answer
46 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 ...
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1answer
27 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 ...
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0answers
16 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 ...
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2answers
34 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 ...
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0answers
11 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 ...
5
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2answers
91 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 ...
1
vote
1answer
41 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 ...