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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4
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2answers
120 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
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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
77 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
164 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
10k 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
102 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
263 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
433 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
193 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, $a \lt t \lt b$. $30=1+29$ $(1+29)^{65}=(1+29)(1+29)^{64}$
4
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2answers
3k 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
94 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
348 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
180 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
394 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
145 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
230 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
710 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
61 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 ...
9
votes
6answers
384 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
125 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
203 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
78 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
129 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
511 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 ...
3
votes
2answers
2k 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
858 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
78 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
634 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
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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
1k 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
346 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
262 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
177 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
664 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
307 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, ...
2
votes
1answer
72 views

Trying to rederive an exponential approximation

So I was reading a paper where the following approximation was made. Note that $p$ is small, $L$ is large, and $pL$ is $O(1)$: ...
1
vote
0answers
85 views

integral with bessel function represented as a series [duplicate]

Possible Duplicate: prove equality with integral and series This integral was my homework question with $p=2$ and $n=1$. I am wondering if one can get the general formula for p, or at least ...
1
vote
2answers
104 views

Limit of exponentials

Why is $n^n (n+m)^{-{\left(n+m\over 2\right)}}(n-m)^{-{\left(n-m\over 2\right)}}$ asymptotically equal to $\exp\left(-{m^2\over 2n}\right)$ as $n,m\to \infty$?
1
vote
1answer
466 views

Using binomial theorem find general formula for the coefficients

Using binomial thaorem (http://en.wikipedia.org/wiki/Binomial_theorem) find the general formula for the coefficients of the expantion: $$ ...
2
votes
1answer
892 views

approximation hypergeometric distribution with binomial

Let $X$ be $\rm{Hypergeometric}(2n,\ell,n)$ and $E(X)=\frac{1}{2} \ell=:\mu$. Is it possible and how to approximate the $q$-th central moment $E(X-\mu)^q$ of the hypergeometric distribution by the ...
2
votes
2answers
104 views

Queries regarding saddle point

I was reading this article in wikipedia related to saddle points. When I came across this line In one dimension, a saddle point is a point which is both a stationary point and a point of ...
3
votes
1answer
94 views

is the approximation of the sum true?

Someone commented under my question Calculation of the moments using Hypergeometric distribution that $$ \sum_{k=0}^l\frac{{l \choose k}{2n-l \choose n-k}(2k-l)^q}{{2n\choose n}}\sim \sum_{k=0}^l ...
0
votes
0answers
197 views

Orthogonal basis for waveform expansion

I have many signals where each signal has a different waveform f(x). One example of such a waveform could be this f(x) sampled at 11 x positions: I am looking for a basis, Bi, for a series ...
2
votes
1answer
168 views

Need help understanding Chebyshev approximation

I have a function $f(x)$ sampled at 11 $x$ positions: I want to approximate the function by a Chebyshev expansion: $$ \ f(x) \simeq \sum\limits_{i=0}^m c_i T_i(y) - \frac{1}{2}c_0,\qquad ...
5
votes
1answer
169 views

Why does $\cos (\pi\cos (\pi \cos (\log (20+\pi)))) \approx -1$

I read on Wikipedia that $$\cos (\pi\cos (\pi \cos (\log (20+\pi)))) \approx -1$$ to a high degree of accuracy. Why is this true? Is this pure coincidence or is there some mathematical ...
3
votes
1answer
102 views

A problem on $C^k$

Can anyone help me solve the following problem ? Let $a,k >0$, $a$ is real and $k$ is integer. Consider the set $S$ of all function $f\in C^k([0,a])$ such that 1) $f(0)=0$ and $f(a)=1$ 2) ...
5
votes
2answers
148 views

Approximation by $C^1$ path of a Lipschitz continuous path

I was wondering if the following equality holds: $$\inf\left\{\int_0^1 G(\gamma(t))|\gamma'(t)|dt, \gamma \in X \cap (\text{Lipschitz})\right\}\stackrel{??}{=}\inf\left\{ \int_0^1 ...