Numerical Series Expansion

Mathematicians, scientists, and engineers often need solutions for difficult to unsolvable equations in order to progress in their work. Fortunately, through various methods of numerical and polynomial series expansions, some of the most problematic equations can be approximated to easily workable forms. Please post all questions regarding the various expansions and series, such as the Taylor and Maclaurin series expansions, the binomial expansion, and the geometric expansion, as well as the rules that govern them, into this category.

440 Questions
Math and Arithmetic
Algebra
Geometry
Numerical Series Expansion

How does an infinite geometric series apply to being pushed on a swing?

Probably the movement on a swing can be approximated by assuming that the magnitude of each swing will be a certain percentage of the previous swing (because of lost energy).

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Numerical Series Expansion

What is rapid convergence?

Rapidity of convergence is a relative concept whose meaning comes from the "comparison test" for convergence. One version of the comparison test says that if (an) is convergent and abs(bn)<K*abs(an) for all n, for some K>0, then (bn) must be convergent too.

99100101
Probability
Numerical Series Expansion
Irrational Numbers

What does numbers in fafi means?

no. 3 = big water or sea

192021
Proofs
Numerical Series Expansion
Complex Numbers

How do you prove Euler's formula?

Euler's formula states that eix = cos(x) + isin(x) where i is the imaginary number and x is any real number.

First, we get the power series of eix using the formula:

ez = Σ∞n=o zn/n! where z = ix. That gives us:

1 + ix + (ix)2/2! + (ix)3/3! + (ix)4/4! + (ix)5/5! + (ix)6/6! + (ix)7/7! + (ix)8/8! + ...

which from the properties of i equals:

1 + ix - x2/2! - ix3/3! + x4/4! + ix5/5! - x6/6! - ix7/7! + x8/8! + ...

which equals:

(1 - x2/2! + x4/4! - x6/6! + x8/8! - ...) + i(x - x3/3! + x5/5! - x7/7! + ...).

These two expressions are equivalent to the Taylor series of cos(x) and sin(x). So, plugging those functions into the expression gives cos(x) + isin(x).

Q.E.D.

Of course, were you to make x = п in Euler's formula, you'd get Euler's identity:

eiπ = -1

It depends on your definitionThe question presumes that someone has already given a definition of the exponential function for complex numbers. But has this definition been given? If so, what is this official definition?

The answer above assumes that the exponential function is defined, for all complex numbers, by its power series. (Or, at least, that someone else has already proved that the power series definition is equivalent to whatever we're taking to be the official definition).

So the answer depends critically on what the definition of the exponential function for complex numbers is.

Suppose you know everything about the real numbers, and you're trying to build up a theory of complex numbers. Most people probably view it this way: Mathematical objects such as complex numbers are out there somewhere, and we have to find them and work out their properties. But mathematicians look at it slightly differently. If we can construct a thing which has all the properties we feel the field of complex numbers should have, then for all practical purposes the field of complex numbers exists. If we can define a function on that field which has all the properties we think exponentiation on the complex numbers should have, then that's as good as proving that complex numbers have exponentials. Mathematicians are comfortable with weird things like complex numbers, not because they have proved that they exist as such, but because they have proved that their existence is consistent with everything else. (Unless everything else is inconsistent, which would be a real pain but is very unlikely.)

So how do we construct this exponential function? One approach would be simply to define it by exp(x+iy) = ex(cos(y)+i.sin(y)). Then the answer to this question would be trivial: exp(iy) = exp(0+iy) = e0(cos(y)+i.sin(y)) = cos(y)+.sin(y). But you'd still have some work to do to prove things like exp(z+w) = exp(z).exp(w). Alternatively, you could define the exponential function by its power series. (There's a theorem that lets you calculate the radius of convergence, and that tells us the radius is infinite, i.e. the power series works everywhere.) Or maybe you could try something like proving that the equation dw/dz = w has a unique solution up to a multiplicative constant, and defining the exponential function to be the solution which satisfies w=1 at z=0.

Let z = cos(x) + i*sin(x)

dz/dx = -sin(x) + i*cos(x)

= i*(i*sin(x) + cos(x))

= i*(cos(x) + i*sin(x))

= i*(z)

therefore dz/dx = iz

(1/z)dz = i dx

INT((1/z)dz = INT(i dx)

ln(z) = ix+c

z = e(ix+c)

Substituting x=0, we get cos(0) + i*sin(0) = e(0i+c)

1+0 = e(0+c)

e(0+c) = 1

therefore c=0

z = eix

eix = cos(x) + i*sin(x)

Substituting x=pi, we get e(i*pi) = -1 + 0

e(i*pi) = -1

e(i*pi) + 1 = 0

qED

697071
Math and Arithmetic
Electronics Engineering
Mathematical Analysis
Numerical Series Expansion

What is the difference between a Fourier series and a Fourier transform?

The Fourier series is an expression of a pattern (such as an electrical waveform or signal) in terms of a group of sine or cosine waves of different frequencies and amplitude. This is the frequency domain.

The Fourier transform is the process or function used to convert from time domain (example: voltage samples over time, as you see on an oscilloscope) to the frequency domain, which you see on a graphic equalizer or spectrum analyzer.

The inverse Fourier transform converts the frequency domain results back to time domain. The use of transforms is not limited to voltages.

656667
Math and Arithmetic
Roman Numerals
Numerical Series Expansion

What number does the big hand point to show quarter past?

The 15, easy

636465
Consumer Electronics
Physics
Numerical Series Expansion
Similarities Between

What is the relationship between frequency and harmonics?

Relationship Between Frequency and HarmonicsThe frequency is the fundamental frequency or the operating frequency, and the harmonics are multiples of that frequency which are generally of less amplitude. It's something that's inherent in non-sinusoidal oscillators, like sawtooth oscillators.

In digital, there's also usually a sinusoidal oscillator which is either built in or external, but the oscillations are converted into square waves to create a clock pulse which negates the harmonics.
There is a little problem in counting the harmonics and the overtones. So you find different statements in the internet. Some are wrong and some are right.
Scroll down to related links and look at "Calculations of Harmonics from Fundamental Frequency".
575859
Math and Arithmetic
Statistics
Numerical Series Expansion

What are the 6 characteristics of a binomial distribution?

  1. Each outcome must be classified as a success (p) or a failure (r),
  2. The probability distribution is discrete.
  3. Each trial is independent and therefore the probability of success and the probability of failure is the same for each trial.
282930
Statistics
Numerical Series Expansion

What is the relationship between the binomial expansion and binomial distribution?

First i will explain the binomial expansion

495051
World Series
Warriors Book Series
Plural Nouns
Numerical Series Expansion

Why was there no 1994 World Series?

There was a strike that ended the baseball season in mid-August in 1994.

484950
Numerical Series Expansion

What number is larger than the trillions?

now i MAY be wrong, but ill go ahead:

Million

Billion

Trillion

Quadrillion

Pentillin

Sectillion

(Don't know the one for 7)

Octillion

And that's all I can figure, hope I was help XP

Edit: It goes by the common scientific naming of multiplicity, just like what is described above, but you don't use the prefixes used for geometry like what is described. You instead use the common prefixes used in naming chemical compounds. Here's the list:

Million

Billion

Trillion

Quadrillion

Quintillion

Sextillion

Septillion

Octillion

Nonillion

Decillion

Undecillion

Duodecillion

and so on and so forth...duodecillion is actually 10,000,000,000,000,000,000,000,000,000,000,000,000,000 so if you need to really count anything more than that you're probably at a molecular level and would simply call it 10 to the 39th power.

:)

474849
C Programming
Numerical Series Expansion

Derive recursion formula for sin by using Taylor's Series?

the Taylor series of sinx

394041
Statistics
Probability
Numerical Series Expansion

For a binomial distribution with n15 as p changes from .50 toward .05 the distribution will become?

with n=15 as fixed, as p=0.5 changes to p=.05 the binmomial distribution will shift to the right.

394041
School Subjects
Florida State University
Essays
Numerical Series Expansion

What is divergence?

The word divergence means separation, parting or moving in different directions.

Examples:

As we have grown older, my brother's and my taste in music has diverged - I prefer classic 60s rock whereas he likes more modern music.

When a laser beam moving in a straight is deflected in some angle, the laser is diverged.

394041
World Series
Mathematicians
Numerical Series Expansion

What rules are used in the World series al or nl?

It follows the rules of the home team.

293031
World Series
Warriors Book Series
Plural Nouns
Numerical Series Expansion

Why was there no 1904 World Series?

Owing to business rivalry between the two leagues, especially in New York, and to personal animosity between Giants manager John McGraw and American League President Ban Johnson, the Giants declined to meet the champions of the "junior" or "minor" league. McGraw even went so far to say that his Giants were already the World Champions since they were the champions of the "only real major league".

232425
English Spelling and Pronunciation
Numerical Series Expansion

How do you spell fourth?

fourth

151617
Mathematical Analysis
Numerical Series Expansion

What properties of a function can be discovered from its Maclaurin series?

Yes. If the Maclaurin expansion of a function locally converges to the function, then you know the function is smooth. In addition, if the residual of the Maclaurin expansion converges to 0, the function is analytic.

293031
Roman Numerals
Numerical Series Expansion

What does XXXIV represent in roman numerals?

34.

212223
C Programming
Numerical Series Expansion

C programming of Runge-Kutta method?

PROGRAM :-

/* Runge Kutta for a set of first order differential equations */

#include

#include

#define N 2 /* number of first order equations */

#define dist 0.1 /* stepsize in t*/

#define MAX 30.0 /* max for t */

FILE *output; /* internal filename */

void runge4(double x, double y[], double step); /* Runge-Kutta function */

double f(double x, double y[], int i); /* function for derivatives */

void main()

{

double t, y[N];

int j;

output=fopen("osc.dat", "w"); /* external filename */

y[0]=1.0; /* initial position */

y[1]=0.0; /* initial velocity */

fprintf(output, "0\t%f\n", y[0]);

for (j=1; j*dist<=MAX ;j++) /* time loop */

{

t=j*dist;

runge4(t, y, dist);

fprintf(output, "%f\t%f\n", t, y[0]);

}

fclose(output);

}

void runge4(double x, double y[], double step)

{

double h=step/2.0, /* the midpoint */

t1[N], t2[N], t3[N], /* temporary storage arrays */

k1[N], k2[N], k3[N],k4[N]; /* for Runge-Kutta */

int i;

for (i=0;i

{

t1[i]=y[i]+0.5*(k1[i]=step*f(x,y,i));

}

for (i=0;i

{

t2[i]=y[i]+0.5*(k2[i]=step*f(x+h, t1, i));

}

for (i=0;i

{

t3[i]=y[i]+ (k3[i]=step*f(x+h, t2, i));

}

for (i=0;i

{

k4[i]= step*f(x+step, t3, i);

}

for (i=0;i

{

y[i]+=(k1[i]+2*k2[i]+2*k3[i]+k4[i])/6.0;

}

}

double f(double x, double y[], int i)

{

if (i==0)

x=y[1]; /* derivative of first equation */

if (i==1)

x= -0.2*y[1]-y[0]; /* derivative of second equation */

return x;

}

252627
Electronics Engineering
Electrical Engineering
The Difference Between
Numerical Series Expansion

What are harmonics?

Harmonics are multiples (thirds, fifths, etc) or divisions of frequencies. In radio, harmonics can be used carry additional signals on a single base frequency. It is the harmonics of an audio frequency that make a musical instrument unique. By damping a string at a half or third/fifth of its length a harmonic is easily heard. The truth is they were always there, the damping of the incident wave just brings them to the fore. Scroll down to related links and look at "Calculations of Harmonics from Fundamental Frequency".

272829
Numerical Series Expansion

Why we use Taylor's series?

why we use tailor series for calculating of error for a test with multiple variables justify it

192021
Statistics
Numerical Series Expansion

When is a binomial distribution cumulative?

When you add together the probabilities of all outcomes in which the number of successes is equal to or fewer than the given number.

123
Math and Arithmetic
Statistics
Numerical Series Expansion

How many experimental outcomes are possible for the binomial and the Poisson distributions?

The binomial distribution is a discrete probability distribution. The number of possible outcomes depends on the number of possible successes in a given trial. For the Poisson distribution there are Infinitely many.

212223
Chennai
Numerical Series Expansion

Expansion Ukg and Lkg?

Lower Kinder Garten & Upper Kinder Garten

91011

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