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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2answers
162 views

approximation formula for the integral

Get an approximation formula for the following integral: $$ \sum_{k=1}^n \left( \frac{1}{35} \right)^{k-1}\int_0^{\frac{\pi}{2}}\cos^{2(n-k)+1}(y) \cdot \sin^{2(k-1)}(y) \, dy $$
2
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1answer
706 views

Trouble Understanding Newton-Raphson Iterative Method

Currently i am reading this page which discusses the newton-raphson method of approximating the roots of an equation. It says given a function $f$ over the reals $x$, and its derivative $f$,we begin ...
4
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1answer
482 views

Hermite Interpolation of $e^x$. Strange behaviour when increasing the number of derivatives at interpolating points.

I am trying to understand Hermite Interpolation. Here is my pedagogical example. I want to approximate $f(x)=e^x$ on the domain $[-1,1]$ using Hermite interpolation. I choose the Chebyshev zeros ...
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1answer
110 views

Approximation of a function and second derivative

While solving some exercise I came up with this problem: Assume that $f \in C^2[0,\infty]$ (i.e. $f$ has continuous second derivative on $[0,\infty)$ and there exists limit in $+\infty$ of $f$) be ...
2
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2answers
329 views

Approximating $|x|$ by a linear combination of $1, \cos x, \sin x, \cos 2x, \sin 2x$

Let $\phi(x) = |x|$ for $x \in (-\pi, \pi)$. Suppose we approximate $\phi(x)$ by a linear combination of the functions $\{1, \cos x, \sin x, \cos 2x, \sin 2x\}$. What linear combination of the form: ...
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0answers
36 views

Approximation of a function with certain restrictions at problematic points

I can't compute a Taylor series of a function like $f(x)=\sqrt{x}$ to some order around $x_0=0$, because the derivative at that point doesn't exist. If I consider the taylor series $Tf$ at any ...
2
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1answer
211 views

Approximate Set Cover Problem by Rounding

Here is the simple algorithm for approximating set cover problem using rounding: Algorithm 14.1 (Set cover via LP-rounding) Find an optimal solution to the LP-relaxation. Pick all sets ...
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0answers
111 views

Approximating sums like $\sum_{j=1}^n\sum_{k=1}^{\lfloor\frac{n}{j}\rfloor}\int_1^{\frac{n}{jk}}dx$

Can anyone tell me how to approximate the following functions? $f_3(n) = \displaystyle\sum_{j=1}^n\sum_{k=1}^{\lfloor\frac{n}{j}\rfloor}\int_1^{\frac{n}{jk}}dx$ $f_4(n) = ...
2
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2answers
106 views

Approximating $x^k e^{-x}$

I want to approximate the function $ f(x) = x^k e^{-x}$ with some finite series. One approach would be to use the power series expansion for $ e^{-x} $. But in that case, the power series would have ...
3
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1answer
209 views

Calculate $E[X]$ using polynomial approximation of CDF

I have a black box called $F(t)$ ($~$($P~(X\le t)~$, $X$ is random variable) with me where I don't have any information on the exact expression of $F(t)$. But if I supply a $t\ge 0$ I will get a value ...
3
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2answers
80 views

Approximation or bound of $\operatorname{Pr}(X<\operatorname{E}(X))$

$X$ is a continuous random variable (we can assume some statistic (e.g., mean and variance) are known, but the distribution is unknown). Consider a probability ...
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1answer
90 views

Please, Let me know this approximation

$$ N_v = 0.5(t^{2}+2t^{6/7})\ln(1+2t^{-8/7})-t^{6/7} \tag{1} $$ $$ N_v =(0.871+0.125\ln t)^2 \tag{2}$$ Eq(2) is the approximated version of Eq(1). Does anyone know how to derive (2) from ...
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1answer
129 views

Berry-Esseen inequality for the event $a<S_n<b$

Suppose that $X_i$ are independent identically distributed with finite variance and $S_n=X_1+\cdots+X_n$. One can use the Central Limit Theorem to estimate (a) $P(S_n \leq b)$ and (b) $P(a<S_n \leq ...
0
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1answer
103 views

the additional term $4Nd$ in Bakhshali approximation for square roots

In the wiki page of square root calculation, I see an additional term $4Nd$ in Bakhshali approximation than in the Taylor series with 3 terms. I do not see how Bakhshali approximation is "equivalent ...
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2answers
135 views

Fitting a function to a polynomial

I have a black box called $F(t)$ with me where I don't have any information on the exact expression of $F(t)$. But if I supply a $t\ge 0$ I will get a value of $F(t)$ from the black box as output. It ...
2
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1answer
967 views

Power series representation of $e^x$ and $e^{-x}$

The power series representation of $e^x = \sum \limits_{k=0}^{\infty} \frac{x^k}{k!}$. Can I use this approximation for $e^{-x} = 1/e^x = 1/\sum \limits_{k=0}^{\infty} \frac{x^k}{k!}$ instead of ...
1
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0answers
75 views

Approximation in the Plane of Constant Curvature

In Euclidean space, There is a classic Theorem claims that: The length of every rectifiable curve can be approximated by sufficiently small straight line segment with ends on the curve. Now, The ...
2
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1answer
253 views

Polynomial approximation of $\chi^2$ distribution pdf

The $\chi^2$ distribution PDF is $$f_{\chi^2}(x;k) = \frac{1}{2^{k/2}\Gamma(k/2)} x^{k/2 - 1} \mathrm{e}^{-x/2} \mathbf{1}_{x \geq 0}$$ I am trying to find a polynomial approximation to this density ...
3
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0answers
246 views

Approximating a system of differential equations as a Bézier curve

I am looking for a general transform to approximate the solution to an n-dimensional system of differential equations and initial conditions as a cubic or quadratic Bézier curve. Sorry if my ...
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1answer
47 views

Approximation of closest k-coloured points?

I'm a working software engineer faced with the following problem: I have a set of points on a 2d plane. Each point can have one of $k$ different colours. I wish to select one point of each colour that ...
4
votes
1answer
109 views

How can we approximate $\sum_{j=0}^n{\sum_{k=0}^j{c^j k^{1/2}}}$ by integrals?

"Difference Equations" by Walter G. Kelley and Allan C. Peterson, 2nd Edition, gives an example on how to approximate $\sum_{k=1}^n{k^{1/2}}$ using integrals and Bernoulli numbers. I'm interested in ...
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1answer
215 views

Solve $\displaystyle\int_{0}^a \left(3^{\frac{1}{3} \left( x^3 - 3x \right) }-1\right)\, dx = 0$ using elementary methods

A friend of mine came upon the following problem. Solve for $a$ the equation $\displaystyle\int_{0}^a \left(3^{\frac{1}{3} \left( x^3 - 3x \right)}- 1\right)\, dx = 0$. By typing the problem ...
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2answers
1k views

Help finding the absolute error with $n$th degree Taylor polynomials

I am trying to estimate the absolute error in approximating $\ln 1.09$ with the $3$rd-order Taylor polynomial centered at $0$. It's been a while since I've taken the Calculus and I'm afraid I need ...
1
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1answer
52 views

Euler's approximation of $m' = -\frac{m}{V}v$

Continuing my previous question from today - I should code a program that finds approximate value of the aforementioned differential equation. The full text of the assignment is: Water purifier with ...
2
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2answers
353 views

Triple Recursion Relation Coefficients

I am reading Atkinson's "An Introduction to Numerical Analysis" and I am trying to understand how a certain equation was reached. It is called the "Triple Recursion Relation" for an orthogonal family ...
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2answers
1k views

Approximating a sum of exponential distribution with a normal distribution

Here is the actual question: $A$ is random variable representing the lifespan of a component. It is an exponential law with an average of 10. Considering a system with $n$ components $A$, what is the ...
2
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1answer
287 views

Euler's method approximation

I'm supposed to write a program for approximating the value of function $y = y(x)$, which is given as: $$y' = \frac{1+y}{1 + x^2}$$ I also know that $y(0) = 0$. I should approximate the value for ...
3
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2answers
605 views

Limit Question with Bernstein Polynomial approximating $f(x) = x^2$

I am reading Atkinson's "An Introduction to Numerical Analysis". I am trying to verify a limit on page 198 involving the Bernstein polynomial approximating $f(x) = x^2$. The statement in the book is ...
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1answer
855 views

Approximate the integral $\int_0^1 \sin(x^2) dx$

I'd like to ask if someone can please give me a little push with this assignment: Approximate the value of the integral $\int_0^1 \sin(x^2) dx$ using only $\mathbb{N}$ numbers and basic operations ...
3
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1answer
318 views

Approximation with complex polynomials on $S^1$ - can it be done?

Can one uniformly approximate a function 'similar' to identity on $S^1$ with complex polynomials? I mean a function like: $f(z)=z \cdot (1+h \cdot \sin(m\cdot Arg(z)))$, for $|h| < 1,\ m \in ...
3
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2answers
433 views

Approximate a positive Sobolev function by positive smooth functions

Here is a problem that I have encountered in PDE book several times. But I have never seen a proof of it. I will be very grateful if someone could give me a proof. Question: Let $B$ be the unit ball ...
3
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1answer
916 views

With Euler's method for differential equations, is it possible to take the limit as $h \to 0$ and get an exact approximation?

I was recently watching a tutorial on Euler's method for approximating differential equations, and the whole time I was thinking "why can't you just take the limit of the step size $h$ as it goes to ...
1
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1answer
201 views

Midpoint Rule, Trapezoidal Rule, etc.: When the number of intervals increases by a factor of $q$, the approximation error decreases by $r(q) =\;$?

I'd like to look at this problem in terms of the definite integral $I = \int_0^5 e^{\sin\sqrt x}dx$, and in terms of the Midpoint Rule. (Then, hopefully, I'll be able to figure out the left-point ...
1
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0answers
146 views

Spline function: Parametric representation of a curve

I have this problem: I have a curve/figure on a sheet of graphpaper. Then I have to select points, read the x,y-values and label them $t_{0} = 1.0$ and $ t_{1}=2.0$ ect. I now have to obtain a table ...
1
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2answers
102 views

Low degree approximation of the polynomial extension of the logical-or function

Let $x\in\{0,1\}^n$ be a binary vector of dimension $n$, and let $OR(x)$ be the "logical or" function (i.e., returns $1$ if at least one of the coordinates is $1$ and otherwise returns $0$). Consider ...
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5answers
3k views

Approximation of $e^{-x}$

Is there a method to mentally evaluate $e^{-x}$ for $x>0$? Just to have an idea when computing probabilities or anything that is an exponential function of some parameters.
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1answer
458 views

How to select a sensible tolerance when making approximations in MATLAB

I'm approximating $\pi$ using a series in MATLAB. I can approximate to within a relative error of $3\times 10^{-10}$ of MATLAB's built-in value. How would I choose a sensible tolerance for my ...
3
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2answers
665 views

How to bound the truncation error for a Taylor polynomial approximation of $\tan(x)$

I am playing with Taylor series! I want to go beyond the basic text book examples ($\sin(x)$, $\cos(x)$, $\exp(x)$, $\ln(x)$, etc.) and try something different to improve my understanding. So I ...
8
votes
4answers
213 views

How could we manually approximate $\sum_{i=1}^{50} i! = 3.1035 \times 10^{64}$?

How could we manually approximate $$\sum_{i=1}^{50} i!$$ to the value $ 3.1035 \times 10^{64}$? I faced this question in my aptitude test,there were four option given,I couldn't solve it during ...
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2answers
4k views

Approximating the logarithm of the binomial coefficient

We know that by using Stirling approximation: $\log n! \approx n \log n$ So how to approximate $\log {m \choose n}$?
9
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1answer
780 views

Tensor products of functions generate dense subspace?

Let $X$ and $Y$ be two spaces in certain category, $F(\cdot)$ a functor associating each space with a function space (with certain topology). Assume that for any $f\in F(X)$ and $g\in F(Y)$, $f\otimes ...
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2answers
2k views

How can I calculate non-integer exponents?

I can calculate the result of $x^y$ provided that $y \in\mathbb{N}, x \neq 0$ using a simple recursive function: $$ f(x,y) = \begin {cases} 1 & y = 0 \\ (x)f(x, y-1) & y > 0 \end ...
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1answer
729 views

A series problem by Knuth

I came across the following problem, known as Knuth's Series which originally was an American Mathematical Monthly problem. Prove that $$\sum_{n=1}^\infty ...
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3answers
672 views

How to do this approximation?

Question: Of the following, which is the best approximation of $$\sqrt{1.5}(266)^{3/2}$$ $$(A)~1,000~~~~(B)~2,700~~~~(C)~3,200~~~~(D)~4,100~~~~(E)~5,300$$ I used $1.5\approx1.44=1.2^2$ and ...
5
votes
4answers
3k views

Simulate a double chance bet with two single bets

If you bet on the result of a soccer match, you usually have three possibilities. You bet 1 - if you think the home team will win X - if you think the match ends in a draw 2 - if you think the away ...
3
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4answers
719 views

Find approximation to $\sin(x)$

How to find the approximation to $\sin(1.58)$ ? By using the Newton's method $$x_{n+1} = x_{n} - \frac{f(x)}{f'(x)}$$ You always will get $0$. Using this method: $f(x+\Delta x) \approx f(x) + ...
12
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9answers
7k views

How to calculate $e^x$ with a standard calculator

Is there a simple method for calculating the $e^x$ ($x\in\mathbb{R}$) with a basic add/subtract/multiply/divide calculator that converges in reasonable time, preferably without having to memorize ...
1
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2answers
183 views

Approximation of a function with polynomials

I have a function of the form $$ f(k)=\frac{1}{a_1-a_{2}k^2e^{-(a_{3}-a_{4}k^2)}};\quad k=0...n$$ I approximated it with Taylor series expansion around $k=\frac{n}{2}$, but the results is not very ...
3
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1answer
79 views

Disc method of approximating volume

There are a couple things I'm unclear on regarding the disc method of approximating shape volume. Given $x=y^2$ and $x=4$, determine the approximate volume by revolving the shape around the line ...
1
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1answer
45 views

approximate the probability of fixed length string segments match

say, i have got 3x'a', 5x'b', 2x'c',4x'd', as char collection. 2 strings are formed, each consists of all the chars given. eg. string A = 'aaabbbbbccdddd' B='abcdabcdabbbdd' so both strings have ...