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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3
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1answer
251 views

What is the proof of the rules of significant figures?

I wanted to know how do we know that the rules that we follow when doing arithmetic with significant figures are correct? Like why when adding or subtracting we keep the same number of decimal places ...
3
votes
3answers
63 views

Approximation for $2^r\ln \frac{2^r}{2^r-r}$

I know the function $$2^r\ln \frac{2^r}{2^r-r}$$ is about linear in $r$, but I need an argument that an undergraduate could follow. Is there a simple way to explain this? I'd be happy with a simple ...
2
votes
3answers
366 views

Simple test if point is above or below sine curve

Is there any simple formula or algorithm for determining if a point lies above or below the sine curve? For instance, if I have a point $(x, y)$, how can I test whether or not $y > \sin(x)$? ...
1
vote
0answers
88 views

Rapidly convergent series for $\sum_{J=0}^{\infty} (2 J + 1) e^{-\beta J(J+1)}$ (rigid rotor)

I need to evaluate this series for arbitrary $\beta > 0$: $ Q = \sum_{J=0}^{\infty} (2 J + 1) e^{-\beta J(J+1)} $ Is it related to a known transcendental function? From the research I did, it ...
2
votes
2answers
725 views

How many bits are in factorial?

I am interested in good integer approximation from below and from above for binary Log(N!). The question and the question provides only a general idea but not exact values. In other words I need ...
1
vote
0answers
198 views

Approximation algorithm for sine function

What is the most common approximation algorithm for sine function which can be implemented with self-organizing maps algorithm? I was thinking of least-squares approximation but I can't find a ...
2
votes
1answer
138 views

Upper bound for $\Gamma(x+y)$

Let $x, y \geq 1$ be two real numbers. I am wondering if one can get an upper bound for $\Gamma(x+y)$ in terms of $\Gamma(x)\Gamma(y)$? Any references or ideas are very appreciated. Thank you.
1
vote
1answer
113 views

Question about piecewise linear paths

I'm having trouble with a concept about piecewise linear paths. I've been looking on the net but I haven't found a definition of it. Consider a finite family of hyperplanes in a finite-dimension real ...
2
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0answers
436 views

Polynomial Interpolation and Error Bound

Problem: Use the Lagrange interpolating polynomial of degree three or less and four digit chopping arithmetic to approximate cos(.750) using the following values. Find an error bound for the ...
1
vote
1answer
62 views

How to approximate a complex linear homogenous recurrence with constant coefficients with a simple one?

Is there some standard way to approximate a complex linear homogenous recurrence with constant coefficients with a simple one? For example, I might want to approximate $$ ...
2
votes
2answers
435 views

Where is the error on this approximation to $\pi$

I came across this website (see also) where the author (a supposedly alternative mathematician) claims to have a better approximation to $\pi$. $$\pi\approx 3.1547…$$ Can someone tell me where is ...
1
vote
1answer
194 views

Quadratic approximation of a cost function with a Taylor expansion

See also http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-832-underactuated-robotics-spring-2009/readings/MIT6_832s09_read_ch12.pdf, page 92. Given an instantaneous cost ...
4
votes
2answers
113 views

Approximation with 1-exponential

How come that $$\left(1-\frac{1}{x}\right)^x \approx e^{-1}\ ?$$ Is there a proof or something to understand this?
0
votes
2answers
80 views

Approximating a solution where the existence is guaranteed by IFT

background: I'm working on a problem that uses the implicit function theorem to show the existence of a solution. I have a continuously differentiable function $f(x,y)=0$ with nonzero Jacobian at a ...
2
votes
0answers
145 views

Generating all positive integers from three operations

This question arose on sci.math, but since almost all the competent mathematicians there have migrated here, I thought I'd give the question a wider audience. Starting with an integer $t>2$, ...
1
vote
1answer
74 views

How to approximate the error of a starting guess?

Given the equation $$ \frac{1}{x^4}+e^{x-100}=10^8 $$ that has one positive root > 1, formulate Newton's method for finding the root. Make one iteration with starting value = 1. Try to make another ...
2
votes
2answers
145 views

If value halves every $5$ years, when will the dollar be worth $1/1,000,000$ its current value?

This was a GRE multiple choice question. At a $15$ percent annual inflation rate, the dollar would decrease by approximately one-half every $5$ years. At this inflation rate, in approximately how ...
7
votes
4answers
8k views

How to convert radicals to decimals without a calculator

How can one convert radicals to decimals(approximate value) when the number is not perfect such as $\sqrt2$, $\sqrt3$, $\sqrt5$, etc. Without the use of calculators.
1
vote
2answers
101 views

Estimating sums

Estimate following sums as the functions of variable $n$: a) $\displaystyle\sum_{i=1}^{n}e^i\ln i$ b) $\displaystyle\sum_{i=1}^{2n}(-1)^i\ln i$ c) $\displaystyle\sum_{i\ge ...
1
vote
0answers
245 views

Bound on euclidean norm

Is it possible to find a suitable lower bound on $$\left|\left(tM+\sum_{k=2}^\infty\frac{(tM)^k}{k!}\right)\cdot b\right|$$ for $M$ as $n \times n$ matrix, $b$ as $1 \times n$ vector and for all $t$ ...
4
votes
1answer
422 views

Can the trigonometric functions be expressed, explained, or proven in terms of arithmetic?

I'm trying to wrap my head around sine, cosine, and tangent. I'm aware that they're commonly defined in high schools as ratios of the various parts of triangles set in the unit circle, but that's not ...
3
votes
3answers
191 views

$t=\frac{30^{65}-29^{65}}{30^{64}-29^{64}}$, find the closest pair of integers, a and b, such that, $a \lt t \lt b$.

$t=\frac{30^{65}-29^{65}}{30^{64}-29^{64}}$ find the closest pair of integers, a and b, such that, $a \lt t \lt b$. $30=1+29$ $(1+29)^{65}=(1+29)(1+29)^{64}$
3
votes
2answers
2k views

Modern formula for calculating Riemann Zeta Function [duplicate]

Possible Duplicate: How to evaluate Riemann Zeta function I have an amateur interest in the Zeta Function. I have read Edward's book on the topic, which is perhaps a little dated. I would ...
3
votes
1answer
90 views

Calculating the divisor, known to be small, of two Stirling approximations of the logarithmic Gamma function without overflows

Earlier, I asked a question on MathOverflow regarding how one might analytically approximate a function of the form: $f(n) = \prod_{i=1}^{n-1} (1-ai)$ for $a \ge 0$, $(ai) < 1$, and $n > 10^5$ ...
1
vote
0answers
285 views

Polynomial approx to the Normal density

I have found several polynomial some approximations to the Normal CDF$^{(1)}$, but my question is: are there good polynomial approximations to the Normal PDF? Thanks $^{(1)}$ For example, some are ...
1
vote
0answers
174 views

Solving or approximating an equation with radicals and arctan function

I have solved a differential equation recently, which left me with this whopper of inverse function to figure out. I know what $c$ is, I just haven't calculated its exact value based on the initial ...
10
votes
2answers
385 views

Summation of divergent series of Euler: $0!-1!+2!-3!+\cdots$

Consider the series $$\sum\limits_{k=0}^\infty (-1)^kk!x^k\in\mathbb{R}[[x]]$$ Let $s_n(x)$ be partial sum. And let $\omega_{k,n}=(k!^2(n-k)!)^{-1}$. My question is: Prove that ...
0
votes
1answer
132 views

Counting Weak Compositions and Approximating Alternating Sum

I have the following problem: "Suppose you have a universe of $N$ distinct objects, and you observe $k$ of them, possibly with repetition. The order in which the objects are observed does not matter. ...
3
votes
1answer
224 views

Laplace's method with unknown exponent.

Given the integral: $$I = \int_0^a{e^{-\lambda g(x)}f(x)dx}$$ Where $g(x)$ and $f(x$) are both low order positive polynomials, and $\lambda \gg 1$, Laplace's method is commonly used to approximate ...
4
votes
1answer
570 views

Why does relative error give number of correct digits?

I learnt that if the relative error is 5*$10^{-s}$ then the number of correct digits the result has $s$. Why is this so? Can you illustrate with an example and/or a proof? Another way to put it ...
0
votes
0answers
60 views

Riemann Siegel formula modification?

$$Z(t)= \sum_{n=1}^{\lfloor\sqrt t/2\pi\rfloor }\frac{\cos(N(t)-t\log p_{n})}{\sqrt n}$$ here $N(t)$ is the smooth part of the zeros and $ p_{n} $ are the primes since $ p_{n} =n\log n $ then $ \log ...
8
votes
6answers
359 views

Approximating $\pi$ with least digits

Do you a digit efficient way to approximate $\pi$? I mean representing many digits of $\pi$ using only a few numeric digits and some sort of equation. Maybe mathematical operations also count as ...
3
votes
1answer
119 views

probability involving matching of discrete shapes on a square grid

Figure F exists on a regular square grid. T transforms F by any combination of horizontal or vertical reflection as well as rotation by 90 or 180 degrees. A larger background grid of X by Y contains ...
0
votes
2answers
95 views

Approximate (but as accurate as it can) location of sound

I quess that this is relatively easy question, but I have been struggling this for a two days now (basicly investigating different formulas) and couldn't find a solution. So, let's think this case: ...
2
votes
1answer
180 views

Polynomial Approximation of an Integral

I require a polynomial $p(x)$ such that $$\left|p(x) - \int_0^x \cos{(t^2)} dt\right| < \frac{1}{10!}$$ for all $x \in [-1, 1]$. I know that I should probably use the fact that if $$m\leq f^{n+1} ...
0
votes
0answers
66 views

Approximating number of nodes expanded by A* search

When searching over a graph expressed as a uniform, 8-connected grid using the A* algorithm, is there any way to give a rough approximate of the number nodes expanded? I appreciate this is a somewhat ...
0
votes
1answer
124 views

Approximation of function on interval

I'm looking for an accurate but as simple as possible approximation of $S(x,\lambda) = \frac{1}{(1-x) [x-(1-\lambda )]}\left((1+\lambda ) \left(\frac{x(1+\lambda)}{1-\lambda ...
2
votes
3answers
2k views

Smooth approximation of absolute value inequalities

Is there an analytic approximation to the inequality: $$\sum_{i=1}^{n} |x_i| \leq \delta ? $$ I would like to replace the above inequality with a smooth inequality that is "valid" in the sense that ...
2
votes
4answers
500 views

Approximate solution for the root of a non-linear function

I have been working with a system which involves computing the roots of functions that look like \begin{equation} e^t (g\cos(\omega t) + b) = c \end{equation} where $t$ is the independent variable ...
2
votes
1answer
1k views

Approximations for the partial sums of exponential series

Though the question here (Partial sums of exponential series - Stack Exchange) is similar, it is more specialized and I rather need a general approximation for an arbitrary partial sum. Essentially, ...
2
votes
3answers
793 views

How to calculate the area of bizarre shapes

I'm looking for an algorithm to calculate the area of various shapes (created out of basic shapes such as circles, rectangles, etc...). There are various possibilities such as the area of 2 circles, 1 ...
1
vote
0answers
73 views

How can I compare two approximants to a bivariate function?

An application I'm working on has required me to find simple approximations to a rather complicated bivariate function $g(x,y)$ that also takes a long time to evaluate on the computer. Through sheer ...
0
votes
1answer
507 views

Significant figures with a plus-minus

A question on my homework asks me to give the amount of significant figures of $2900±100$. Would this be one, two, or both?
1
vote
1answer
50 views

Approximation when $|a(t)|\ll b$

If $|a(t)|\ll b$ is it alright to take $a\left({a\cdot \dot{a} \over b^2}\right)$ as $0$? Would the following argument make sense? I know that we can take $\left({a\cdot a \over b^2}\right)$ as $0$ ...
1
vote
1answer
989 views

Approximating Coins Flips Problem

Approximate the probability of getting 500 heads out of a 1000 coin flip of unbiased coins to be within 5% of its true value (without the use of a calculator). I know that an exact probability ...
3
votes
4answers
338 views

What is the correct value of $\pi$

I have seen that: $\pi = 22/7$ $\pi = 3.14\ldots$ $\pi = 17 - \sqrt{192}$. $22/7 \gt \pi$ $22/7 \lt\pi$ My brain storming doubt was, is A = B? Is B = C? Is C= A = B? how D and E are correct or ...
3
votes
3answers
258 views

How to approximate $\sum_{k=1}^n k!$ using Stirling's formula?

How to find summation of the first $n$ factorials, $$1! + 2! + \cdots + n!$$ I know there's no direct formula, but how can it be estimated using Stirling's formula? Another question : Why can't ...
0
votes
1answer
153 views

errors, random and approximations

Hello good evening all! If a reading is reported as R = 200.045 + 0.001 or 200.045 - 0.001 Ohm. Does +0.001 or -0.001 Ohm represents a systematic or random error? Thanking you.
7
votes
2answers
616 views

Approximate $\int_a^b \frac{1}{\sqrt{2 \pi \sigma^2}}e^{-(x-\mu)^2/2 \sigma^2}\log(1+e^{-x}) \ \ dx $

I am trying to find an approximation to $$ I = \int_a^b \frac{1}{\sqrt{2 \pi \sigma^2}}e^{-(x-\mu)^2/2 \sigma^2}\log(1+e^{-x}) \ \ dx. $$ My attempt is as follows: $$ \begin{align} I &= \int_a^b ...
0
votes
1answer
250 views

calculate surface normal with random sampling of points

Given a surface in $R^3$ and a point P on the surface, I want to calculate the surface normal in this point, the vector that is perpendicular to the surface. However, I do not know the whole surface, ...