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
1answer
8k views

Midpoint approximation over/under estimation

So left handed approximation underestimates the area under a increasing curve and over estimates for decreasing curves. And right handed approximation overestimates for increasing curves and ...
1
vote
1answer
57 views

Taylor polynomial approximation (Little $o$)

Suppose $(a_1,...,a_n) \in {\mathbb{R}_{*}^{+}}^n$, how can I prove that using the little $o$ (Taylor polynomial approximation): $\displaystyle\lim _{ x\to +\infty }{ \left( \cfrac { \sum _{ i=1 }^{ ...
2
votes
1answer
148 views

Chebyshev coefficients from a polynomial

Is there an efficient algorithm for finding the coefficients in a Chebyshev basis of a polynomial? That is, given the set of $a_k$ such that: $p_n(x) = \sum_{k=0}^N a_k x^k $ Find the set of $c_k$ ...
2
votes
3answers
78 views

Factoring/approximating an apparently simple formula

Does anyone know if the following formula can be factorized or approximated: $a^3 + b^3 + c^3 + a^2b + ab^2 + a^2c + ac^2 + b^2c + bc^2 + abc$ It looks a lot like $(a + b + c)^3$, except for the ...
3
votes
1answer
62 views

Error of apprximation

Have anyone read book "Paul Glasserman Monte Carlo MIFE", it's good, but i'm stuck in chapter 6 page 341 let $$ dX_t=a(X_t)dt+b(X_t)\,dW_t $$ they said that $$ \int_{t}^{t+\Delta t}a(X_{u}) \, ...
0
votes
2answers
69 views

Confusion regarding an alleged hyperbola

While studying a chapter called price elasticity of demand in my economics course, I have been presented with something called a unit elasticity curve (some sort of a hyperbola), which has supposed ...
0
votes
5answers
894 views

linear or quadratic approximation for exp(-x) for large x

Is there any linear or quadratic approximation of $exp(-x)$ where $0<x<L$ ? $L$ is large, may be 40 (say) i.e. $x$ is not close to zero.
1
vote
1answer
264 views

Showing (1 - polynomial fraction) raised to a polynomial power is a negligible function

Let $P(k)$ and $Q(k)$ be two polynomials ($k>0$). Let $\mathrm{neg}(k)$ be a negligible function for sufficiently large $k$ (see Appendix on question for definition). Does someone know how to show ...
1
vote
0answers
33 views

Iterative approximation of non-constant values in linear equation

The issue regards an algorithm for iterative approximation of unknown transaction values. For each iteration (each day), we are give the total revenue of all transactions for that day, and we have the ...
2
votes
1answer
115 views

Approximating a Gaussian Integral: Can you do better?

I have attempted to approximate this Gaussian: $$ I =\int_{0}^{\lambda}dx\left(r+x\right)\exp\left(-\rho\left(ax^{1/2}+bx^{3/2}\right)\right)\ $$ using $$ I ...
0
votes
1answer
1k 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 ...
7
votes
4answers
513 views

Evaluating the precision in the calculation of $\mathrm{e}$

I'm calculating $\mathrm{e}$ using a computer like this: $$ \mathrm{e} \approx \sum\limits_{i=0}^n {1\over i!} $$ I'm storing it as a rational number. I was wondering, if I write down my rational ...
2
votes
1answer
2k views

Find derivation (dB/decade) for given amplitude characteristic of low pass filter [Hz, -]

I am trying to find derivation (differential attenuation) for frequency's 600 and 2000 Hz for given amplitude characteristic of low pass filter, which look like this: I assume, that I should ...
1
vote
1answer
72 views

Determining the surface with given polynomial borders

Let's say we'd like to guess the shape (I'm not sure the word 'approximate' is appropriate here) of some surface when we are given its borders via third order polynomials (i.e. we are given their ...
0
votes
3answers
59 views

I want to show that $\left | f(3)-P(3) \right |\leq \frac1{1000}$ without a calculator.

Let $f(x)=\ln x$. Then the Taylor polynomial of degree 2 at $x=e$ is $P(x)=1+\frac1e(x-e)-\frac1{2e^2}(x-e)^2$ I want to show that $\left | f(3)-P(3) \right |\leq \frac1{1000}$ without a calculator. ...
0
votes
2answers
55 views

Help evaluating or approximating this integral

For a thermodynamics project I'm working on, I need to evaluate this integral: $\int \frac{(a-bx)(x-c)^d}{x^3}dx$, where $a,b,c,$ and $d$ are all positive constants. I tried evaluating it on ...
0
votes
1answer
64 views

Newton Raphson Method: approximating root

How do we start from approximating a root using this technique? I know of two, viz - a table of x vs f(x), and see where f(x) changes sign - plot a graph, and see where the graph cuts the x axis But ...
1
vote
1answer
149 views

Normal approximation for binomial distribution isn't giving correct result, z score comes out 0

I'm trying to use the normal distribution to calculate approximate values for the (cumulative) binomial distribution with large values (since it's impractical to evaluate the factorials). I'm very ...
1
vote
2answers
46 views

Finding a line of approximation using the normal equations for $A\vec{x}=\vec{b}$

To find the line $y=ax+b$ that best approximates the data points $\{(-2,3),(0,5),(1,7)\}$ I need to use the equation $$A\vec{x}=\vec{b}\ \ \ (\mbox{where}\ \vec{x}=\left({a\atop b}\right))$$ Then ...
0
votes
1answer
47 views

Check my short proof - asymptotic approximation, which function is bigger

The goal of this exercise is to show that $\ln(n+1)-\ln(n) = O(\frac{1}{n})$ what I did is: I used the fact that if $f=O(g)$ then $\frac{f}{g}=O(1)$. $\ln(n+1)-\ln(n)=\ln(\frac{n+1}{n}) = \ln(O(1))$ ...
3
votes
4answers
362 views

Using $(1+x)^k \approx 1+kx$ to approximate?

Use the approximation $(1+x)^k \approx 1+kx$ to estimate $(1.0003)^{26}$ and $\sqrt[4]{1.006}$. I know how to solve this step-by-step, but I don't understand what I'm doing exactly: why does ...
4
votes
1answer
639 views

Newton-Raphson for reciprocal square root

I have a question about using Newton-Raphson to refine a guess of the reciprocal square root function. The reciprocal square root of $a$ is the number $x$ which satisfies the following equation: ...
1
vote
3answers
36 views

How to include the fee of a sell order in itself.

Sorry for the poor title, I am sure there is a name for this problem (and an easy solution) I have an account balance of \$1000. When I want to buy some EUR, I need to sell USD with a fee of 1%. That ...
1
vote
0answers
337 views

Approximate convolution of independent Beta variates?

Is there a way to approximate the convolution of Beta variables? Specifically, I am trying to find an approximation to $g(x_0)$: $$g(x_0) = \int \delta(x_0-\sum_{i=1}^{n} a_i x_i) \prod_{i=1}^{n} ...
0
votes
1answer
193 views

Find an upper bound on the absolute error of 3.141 as an approximation to π

Q: Find an upper bound on the absolute error of 3.141 as an approximation to π I have no idea what to do... :( What I know: absolute error = real value - approximate value Help :)
17
votes
3answers
421 views

Approximating $1/z$ by polynomials

Let $C=\{\mathrm e^{\mathrm it}, 0\le t\le 3\pi/2\}$ and $f(z)=1/z$. By Runge's theorem, there is a sequence of polynomials $p_n(z)$ such that $$\lim_n p_n(z)=f(z)$$ uniformly on $C$. Does anyone ...
4
votes
2answers
122 views

How to approximate $n \int_{0}^{1} [1 - x^m ]^n x^m dx $ near infinity?

I have a hypothesis that if: $$ I_{n,m} := n \int_{0}^{1} [1 - x^m ]^n x^m dx $$ where $m,n \in \mathbb{N}$ then $$ \lim_{n \rightarrow \infty} \frac{I_{n,m}}{n^{-\frac{1}{m} } } = c_m $$ But I ...
2
votes
1answer
99 views

Approximating $\tanh(B\sqrt{A} )$ for small $A$ and arbitrary $B$ by correcting for asymptotic behaviour of $\tanh$

I would like to approximate a function containing terms of the form $\tanh( B\sqrt{A})$ for small $A$. I have tried doing a Taylor series, but I consistently find that it is not only $A$ that has to ...
1
vote
1answer
70 views

How do I show that these two definite integrals are approximately equal

Consider the following two integrals: $I_1=\int\limits_{0}^{1/x}G(s)\ ds$ $I_2=\Big|\int\limits_0^\infty G(s)\exp[-ixs]\ ds\Big|$ where $G(s)$ is a monotonically decreasing positive function, ...
1
vote
1answer
30 views

How to determine coefficients of $p(x) = x^6$ with the Chebyshev processing

I want to calculate the coefficients of $p(x) = x^6$ with the Chebyshev processing. How to do that? Following question would be, how to estimate the error in $[-1,1]$, if i only use terms until ...
2
votes
1answer
45 views

Asymptotic behaviour of the area of a 2-dimensional flat subset of $\mathbb{R}^3$

I am interested by the area of the $2$-dimensional flat subset of $\mathbb{R}^3$ defined by the following equations with one parameter $t>1$: $x,y,z>0$ (positive octant) $x+y+z=t$ (hyperplane ...
2
votes
0answers
344 views

Real approximation to the maximum using Laplace's method integral

The Laplace's Method states that under some conditions, it holds that: $ \sqrt{\frac{2\pi}{M(-g''(x_0))}} h(x_0) e^{M g(x_0)} \approx \int_a^b\! h(x) e^{M g(x)}\, dx \text { as } M\to\infty$ Where ...
1
vote
4answers
142 views

Approximation of $\frac{1+a}{1+b}$

I've found the following assertion on an economics book: For $r$ and $g$ small enough, $\frac{1+r}{1+g}\approx 1+r-g$ (where $r$ is the interest rate and $g$ is the growth rate of the economy) ...
1
vote
1answer
50 views

How to find a sequence by its limit?

Is there any way to construct non-trivial sequence by its limit? Something like $\begin{cases} a_1=2 & \\ a_{n+1}=\dfrac1{2}\left(a_n+\dfrac2{a_n}\right) \end{cases}$for $\sqrt2$. I'm especially ...
3
votes
2answers
116 views

A curious partitions coincidence $\sum_{n=0}^\infty P(n) q^{n+1}$?

Given the partition function $P(n)$ and let $q_k=e^{-k\pi/5}$. What is the reason why, $$\sum_{n=0}^\infty P(n) q_2^{n+1}\approx\frac{1}{\sqrt{5}}\tag{1}$$ $$\sum_{n=0}^\infty P(n) ...
0
votes
1answer
104 views

How do I know that this is the density of the Chebyshev Points?

By knowing that a discrete distribution of points go asymptotically to the density: $\displaystyle p(x)= \frac{1}{\pi \sqrt{1-x²}}$ in $[-1, 1]$ I am able to conclude that interpolating at those ...
1
vote
1answer
157 views

Minimizing the $L^2$ error when approximating with trigonometric polynomials

I want to find approximations ${\rm g}_{n}\left(x\right) \in T_{n}$ of $\,\,{\rm f}\left(x\right)$, so that the error $$ \left\vert\left\vert\,{\rm f} - {\rm g}_{n}\,\right\vert\right\vert^{2} = ...
0
votes
3answers
303 views

Series Expansion of $\arcsin\left(\frac{a}{a+x}\right)$

Can anyone think of a good approximation to: $$ \arcsin\left(\frac{a}{a+x}\right)\ $$ accurate at $x = 0$? The Taylor series is not available...perhaps some other kind of method?
1
vote
1answer
198 views

Regularized distance function on Riemannian manifolds

Suppose $(M,g)$ is a complete Riemannian manifold. $p\in M$ is a fixed point. $d_{p}(X)$ is the distance function defined by $p$ on M (i.e., $d_p(x)$=the distance between $p$ and $x$). Let ...
0
votes
1answer
47 views

Affine approximation of countinuous functions?

Is it true that any real continuous function can be approximated by a piecewise affine function? If true, can you suggest a link or something related to the question? Thanks
3
votes
2answers
84 views

Approximating $e^{-x}$

So we all know that the Taylor series for $e^x$ is $1 + x + \frac{x^2}{2} + \frac{x^3}{6}+\ldots$ Similarly for $e^{-x}$ it comes comes out to be $1 - x + \frac{x^2}{2} - \frac{x^3}{6}+\ldots$ Now for ...
2
votes
6answers
4k views

How to calculate square root or cube root? [duplicate]

I was reading Richard Feynman biography when I read that one time he was able to calculate the cube root of large number in his brain by just using simple facts of everyday life. So my question is ...
1
vote
0answers
21 views

Can $u\in W^{1,2}_0$, $\Delta u\in L^2$, $u\geq 0$ be approximated by a sequence smooth function $u_k\geq 0$

Assume that $\Omega\in R^3$ is a bounded Lipschitz domain. $u\in W^{1,2}_{0}$, $\Delta u\in L^2$, $u\geq 0$. Is it possible to approximate u by a sequence of nonnegative smooth functions $u_k$, ...
4
votes
1answer
401 views

Approximate a complex measurable function pointwisely almost everywhere by polynomials

This is Exercise 13.12 in Rudin's Real and Complex Analysis: Let $f$ be a complex-valued measurable function defined in $\mathbb{C}$. Then there is a sequence of polynomials $P_n$ such that ...
3
votes
2answers
297 views

Sums of central binomial coefficients

Are there closed forms for $$\sum^n_{i=0} \binom{2i}{i}$$ and $$\sum^n_{i=0} \binom{2i}{i}^2$$? Also, how can these sums be approximated?
4
votes
1answer
181 views

Is it possible to approximate all angles with certain pythagorean triples?

With sticks $a,b$ and $c$ of length $3,4$ and $5$, you able to draw a right (tri)angle. But are also able to construct an angle $\cos\alpha=\frac35, \alpha=\arccos(\frac35)=$$53.13010...^°$. Is it ...
1
vote
1answer
35 views

Showing that $\lim\limits _{n\to\infty}{n \choose \left\lceil \frac{n}{2}\right\rceil }\cdot2^{-n}=0$ but the matching series does not converge

I want to show that: $$\lim\limits _{n\to\infty}{n \choose \left\lceil \frac{n}{2}\right\rceil }\cdot2^{-n}=0 $$ And also $${\displaystyle \sum_{n=1}^{\infty}{n \choose \left\lceil ...
0
votes
1answer
214 views

Variational calculus applied to the strain energy functional in solid mechanics

The question is basically about when to apply the variational operator... Given the general functional representing the strain energy of a solid under a given stress state $\sigma$ and strain state ...
0
votes
1answer
44 views

normal as approximation to binomial

Among 784 checks, 479 had amounts with leading digits of 5, but checks issued in the normal course of honest transactions were expected to have 7.9% of the checks with amounts having leading digits of ...
0
votes
1answer
188 views

Intuitive proof of interpolation polynomial existence

Problem: Given a set of $n+1$ data points ($x_i, y_i$) where no two $x_i$ are the same, one is looking for a polynomial $p$ of degree at most $n$ with the property $p(x_i) = y_i$ for all $i∈ [0, n ...