Mathematics Series

11/24/2016

   
   
   
   
 

 

                                                                                                                

e is an important mathematical constant that is the base of the natural logarithm. It is approximately equal to 2.71828 and is the limit of (1 + 1/n)n as n approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series:

e =  \displaystyle\sum\limits_{n = 0}^{ \infty} \dfrac{1}{n!} = 1 + \frac{1}{1} + \frac{1}{1\cdot 2} + \frac{1}{1\cdot 2\cdot 3} + \cdots

In 1737 Euler proved that e is irrational.  In 1873 Hermite proved that e was transcendental.

It is not known if pi+e or pi/e is irrational.

The number e occurs naturally in connection with many problems involving asymptotics. A prominent example is Stirling's formula for the asymptotics of the factorial function, in which both the numbers e and π enter:

n! \sim \sqrt{2\pi n}\, \left(\frac{n}{e}\right)^n.

A particular consequence of this is

e = \lim_{n\to\infty} \frac{n}{\sqrt[n]{n!}}.
 
Euler's formula: e^{ix} = \cos x + i\sin x,\,\!
Euler's formula with x = π is Euler's identity: e^{i\pi} + 1 = 0\,\!
Logarithmic formula: \ln (-1) = i\pi.\,\!
de Moivre's formula.: (\cos x + i\sin x)^n = \left(e^{ix}\right)^n = e^{inx} = \cos (nx) + i \sin (nx),

e is the unique real number such that the value of the derivative (slope of the tangent line) of the function f(x) = ex at the point x = 0 is equal to 1. The function ex so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base e.

The number e is of eminent importance in mathematics, alongside 0, 1, π and i. All five of these numbers play important and recurring roles across mathematics, and are the five constants appearing in one formulation of Euler's identity. Like the constant π, e is irrational: it is not a ratio of integers; and it is transcendental: it is not a root of any non-zero polynomial with rational coefficients. The numerical value of e truncated to 50 decimal places is 2.71828182845904523536028747135266249775724709369995...

e and compound interest 

Let P be the principal (initial investment), rbe the annual compounded rate, i^((n))the "nominal rate," nbe the number of times interest is compounded per year (i.e., the year is divided into nconversion periods), and tbe the number of years (the "term"). The interest rate per conversion period is then

 r=(i^((n)))/n.

If interest is compounded ntimes at an annual rate of r(where, for example, 10% corresponds to r=0.10), then the effective rate over 1/nthe time (what an investor would earn if he did not redeposit his interest after each compounding) is

 (1+r)^(1/n).

The total amount of holdings A after a time t when interest is re-invested is then

 A=P(1+(i^((n)))/n)^(nt)=P(1+r)^(nt).

Note that even if interest is compounded continuously, the return is still finite since

 lim_(n->infty)(1+1/n)^n=e,

where e is the base of the natural logarithm.

The time required for a given principal to double (assuming n=1 conversion period) is given by solving

 2P=P(1+r)^t, or

 t=(ln2)/(ln(1+r)),

where ln is the natural logarithm. This function can be approximated by the so-called rule of 72:

 t approx (0.72)/r.

 

π is a mathematical constant, the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. It is an irrational number. Like all irrational numbers, π cannot be represented as a common fraction. But every irrational number, including π, can be represented by an infinite series of nested fractions, called a continued fraction:


\pi=3+\textstyle \frac{1}{7+\textstyle \frac{1}{15+\textstyle \frac{1}{1+\textstyle \frac{1}{292+\textstyle \frac{1}{1+\textstyle \frac{1}{1+\textstyle \frac{1}{1+\ddots}}}}}}}

When Euler solved the Basel problem in 1735, finding the exact value of the sum of the reciprocal squares, he established a connection between π and the prime numbers that later contributed to the development and study of the Riemann zeta function:

 \frac{\pi^2}{6} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots

Swiss scientist Johann Heinrich Lambert in 1761 proved that π is irrational, meaning it is not equal to the quotient of any two whole number Lambert's proof exploited a continued-fraction representation of the tangent function. French mathematician Adrien-Marie Legendre proved in 1794 that π2 is also irrational. In 1882, German mathematician Ferdinand von Lindemann proved that π is transcendental, confirming a conjecture made by both Legendre and Euler.

Other series involving π are found at: http://mathworld.wolfram.com/PiFormulas.html  Here a very few:

pi/4 = sum_(k=1)^(infty)((-1)^(k+1))/(2k-1) = 1-1/3+1/5-...   discovered by Leibnitz in 1673.

 pi=sum_(k=1)^infty(3^k-1)/(4^k)zeta(k+1),

 pi=sum_(k=0)^infty(2(-1)^k3^(1/2-k))/(2k+1)

1/4pisqrt(2) = sum_(k=1)^(infty)[((-1)^(k+1))/(4k-1)+((-1)^(k+1))/(4k-3)] = 1+1/3-1/5-1/7+1/9+1/(11)-...

1/4(pi-3) = sum_(k=1)^(infty)((-1)^(k+1))/(2k(2k+1)(2k+2)) = 1/(2·3·4)-1/(4·5·6)+1/(6·7·8)-...

1/6pi^2 = sum_(k=1)^(infty)1/(k^2)= 1+1/4+1/9+1/(16)+1/(25)+...

1/8pi^2 = sum_(k=1)^(infty)1/((2k-1)^2)= 1+1/(3^2)+1/(5^2)+1/(7^2)+...

pi = (3sqrt(3))/4+24(1/(12)-1/(5·2^5)-1/(28·2^7)-1/(72·2^9)-...), was determined by Isaac Newton in 1666.

pi/2 =1/2sum_(n=0)^(infty)((n!)^22^(n+1))/((2n+1)!)=sum_(n=0)^(infty)(n!)/((2n+1)!!) = 1+1/3+(1·2)/(3·5)+(1·2·3)/(3·5·7)+... = 1+1/3(1+2/5(1+3/7(1+4/9(1+...))))

 1/9sqrt(3)pi=1/2sum_(i=0)^infty((i!)^2)/((2i+1)!),

 pi/(5sqrt(phi+2))=1/2sum_(i=0)^infty((i!)^2)/(phi^(2i+1)(2i+1)!), where phi is the golden ratio.

 pi=3+1/(60)(8+(2·3)/(7·8·3)(13+(3·5)/(10·11·3)(18+(4·7)/(13·14·3)(23+...)))).

 pi=sum_(n=0)^infty(4/(8n+1)-2/(8n+4)-1/(8n+5)-1/(8n+6))(1/(16))^n.

The first 16 million digits of pi have passed all the tests of being random.

Bellard's formula, as used by PiHex, the now-completed distributed computing project, is used to calculate the nth digit of π in base 2. It is a faster version (about 43% faster[1]) of the Bailey–Borwein–Plouffe formula.  It was discovered by Fabrice Bellard in 1997.

See http://en.wikipedia.org/wiki/Category:Pi

The Feynman point is a sequence of six 9s that begins at the 762nd decimal place of the decimal representation of π. It is named after physicist Richard Feynman, who once stated during a lecture he would like to memorize the digits of π until that point, so he could recite them and quip "nine nine nine nine nine nine and so on", suggesting, in a tongue-in-cheek manner, that π is rational.

Which is a better fit: a round peg in a square hole or a square peg in a round hole?  The ratios are:  π/2  and 2/π .As π squared is greater than 8, the round peg is a better fit and fills more of the square hole.

Buffon's Needle Problem involves pi. . See http://mste.illinois.edu/reese/buffon/buffon.html and http://www.mathsisfun.com/activity/buffons-needle.html

Sum of 1 - 2 + 3 - 4 + 5 - 6  ... =
 Summation from n equals 1 to m of the series n * (-1)^(n-1)
http://en.wikipedia.org/wiki/1_%E2%88%92_2_%2B_3_%E2%88%92_4_%2B_%C2%B7_%C2%B7_%C2%B7
 

In a 1749 report, Leonhard Euler admits that the series diverges but prepares to sum it anyway:

...when it is said that the sum of this series 1−2+3−4+5−6 etc. is 14, that must appear paradoxical. For by adding 100 terms of this series, we get −50, however, the sum of 101 terms gives +51, which is quite different from 14 and becomes still greater when one increases the number of terms. But I have already noticed at a previous time, that it is necessary to give to the word sum a more extended meaning....

Euler proposed a generalization of the word "sum" several times; see Euler on infinite series. In the case of 1 − 2 + 3 − 4 + ..., his ideas are similar to what is now known as Abel summation:

...it is no more doubtful that the sum of this series 1−2+3−4+5 + etc. is 14; since it arises from the expansion of the formula 1(1+1)2, whose value is incontestably 14. The idea becomes clearer by considering the general series 1 − 2x + 3x2 − 4x3 + 5x4 − 6x5 + &c. that arises while expanding the expression 1(1+x)2, which this series is indeed equal to after we set x = 1.

There are many ways to see that, at least for absolute values |x| < 1, Euler is right in that

1-2x+3x^2-4x^3+\cdots = \frac{1}{(1+x)^2}.

 

One can take the Taylor expansion of the right-hand side, or apply the formal long division process for polynomials. Starting from the left-hand side, one can follow the general heuristics above and try multiplying by (1+x) twice or squaring the geometric series 1 − x + x2 − .... Euler also seems to suggest differentiating the latter series term by term.

In the modern view, the series 1 − 2x + 3x2 − 4x3 + ... does not define a function at x = 1, so that value cannot simply be substituted into the resulting expression. Since the function is defined for all |x| < 1, one can still take the limit as x approaches 1, and this is the definition of the Abel sum:

\lim_{x\rightarrow 1^{-}}\sum_{n=1}^\infty n(-x)^{n-1} = \lim_{x\rightarrow 1^{-}}\frac{1}{(1+x)^2} = \frac14.

Convergent series

A series  ∑an  is said to 'converge' or to 'be convergent' when the sequence SN of partial sums has a finite limit. If the limit of SN is infinite or does not exist, the series is said to diverge. When the limit of partial sums exists, it is called the sum of the series

\sum_{n=0}^\infty a_n = \lim_{N\to\infty} S_N = \lim_{N\to\infty} \sum_{n=0}^N a_n.

An easy way that an infinite series can converge is if all the an are zero for n sufficiently large. Such a series can be identified with a finite sum, so it is only infinite in a trivial sense.

Consider the example

 1 + \frac{1}{2}+ \frac{1}{4}+ \frac{1}{8}+\cdots+ \frac{1}{2^n}+\cdots.

It is possible to "visualize" its convergence on the real number line: we can imagine a line of length 2, with successive segments marked off of lengths 1, ½, ¼, etc. There is always room to mark the next segment, because the amount of line remaining is always the same as the last segment marked: when we have marked off ½, we still have a piece of length ½ unmarked, so we can certainly mark the next ¼. This argument does not prove that the sum is equal to 2 (although it is), but it does prove that it is at most 2. In other words, the series has an upper bound. Given that the series converges, proving that it is equal to 2 requires only elementary algebra. If the series is denoted S, it can be seen that

S/2 = \frac{1+ \frac{1}{2}+ \frac{1}{4}+ \frac{1}{8}+\cdots}{2} = \frac{1}{2}+ \frac{1}{4}+ \frac{1}{8}+ \frac{1}{16} +\cdots.

Therefore,

S-S/2 = 1 \Rightarrow S = 2.\,\!

Twin primes are (3,5) (5,7) (11,13) (17,19) etc

The infinite series (1/3 +1/5) + (1/5 + 1/7) + (1'11 + 1/13) + converges to around 1.902160...  This was discovered by the Norwegian mathematician Viggo Brun (1885 - 1978).  It is known as Brun's constant. 

Euler must be regarded as the first master of the theory of infinite series. He created it and was by far its greatest master.  Before Euler entered the mathematical scene there was no systematic theory in dealing with infinite series. Also, most of the series considered had only positive terms.

Archimedes examined the geometric series with first term a1 = a and common ratio r, the sum of the first n terms is given by:

sum, from i = 1 to n, of a-sub-i  is equal to (a) [ (1 - r^n) / (1 - r) ]

 In the special case that | r | < 1, the infinite sum exists and has the following value:

sum, from i = 1 to infinity, of a-sub-i  is equal to  a/(1 - r)

The infinite sum of the power series where a = 1 and r = 1/4  equals 4/3.

 The Leibniz  formula for π, named after Gottfried Leibniz, states that:

 
\sum_{n = 0}^\infty \frac{(-1)^{n}}{2n+1} \;\;=\;\; 1 \,-\, \frac{1}{3} \,+\, \frac{1}{5} \,-\, \frac{1}{7} \,+\, \cdots \;\;=\;\; \frac{\pi}{4}.

The series


\sum_{n = 1}^\infty \frac{(-1)^{n + 1}}{n} \;=\; 1 \,-\, \frac{1}{2} \,+\, \frac{1}{3} \,-\, \frac{1}{4} \,+\, \frac{1}{5} \,-\, \cdots

is known as the alternating harmonic series. This series converges and the sum is equal to the natural logarithm of 2  which equals 0.69314718056...  

1 \,-\, \frac{1}{2} \,+\, \frac{1}{3} \,-\, \frac{1}{4} \,+\, \frac{1}{5} \,-\, \cdots \;=\; \ln 2.

 Another series that presents itself as being similar is the \the sum of the squares of reciprocals of the natural numbers". This was the famous Basel problem  posed by Pietro Mengoli in 1644 and solved by Leonhard Euler in 1735. Euler's solution brought him immediate fame when he was twenty-eight. Euler generalized the problem considerably, and his ideas were taken up years later by Bernhard Riemann in his seminal 1859 paper On the Number of Primes Less Than a Given Magnitude, in which he defined his zeta function and proved its basic properties. The problem is named after Basel, hometown of Euler as well as of the Bernoulli family who unsuccessfully attacked the Basel problem.  The Basel problem asks for the precise summation of the reciprocals of the squares of the natural numbers, i.e. the precise sum of the infinite series:


\sum_{n=1}^\infin \frac{1}{n^2} =
\lim_{n \to +\infty}\left(\frac{1}{1^2} + \frac{1}{2^2} + \cdots + \frac{1}{n^2}\right).

The series is approximately equal to 1.644934...  The Basel problem asks for the exact sum of this series as well as a proof that this sum is correct. Euler found the exact sum to be π2/6 and announced this discovery in 1735. His arguments were based on manipulations that were not justified at the time, and it was not until 1741 that he was able to produce a truly rigorous proof.

Euler used same process to find the precise summation of the fourth powers of the reciprocals of the natural numbers (1 + 1/16 + 1/81 +  1/256 + ... ) which is π4/90 . No one knows the sum of the inverted cubes of the positive integers. You could become very famous if you discover that value.

The Riemann zeta function or Euler–Riemann zeta function ζ(s) is a function of a complex variable s = σ + it. (The notation with s, σ, and t is traditionally used in the study of the ζ-function, following Riemann.)

The following infinite series converges for all complex numbers s with real part greater than 1, and defines ζ(s) in this case:


\zeta(s) =
\sum_{n=1}^\infty n^{-s} =
\frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots \;\;\;\;\;\;\; \sigma = \mathfrak{R}(s) > 1.
\!

 This function, as a function of a real argument, was introduced and studied by Leonhard Euler in the first half of the eighteenth century without using complex analysis, which was not available at that time. Bernhard Riemann in his article "On the Number of Primes Less Than a Given Magnitude" published in 1859 extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation and established a relation between its zeros and the distribution of prime numbers.

Leonhard Euler considered the above series in 1740 for positive integer values of s, and later Chebyshev extended the definition to real s > 1.

The above series is a prototypical Dirichlet series that converges absolutely to an analytic function for s such that σ > 1 and diverges for all other values of s. Riemann showed that the function defined by the series on the half-plane of convergence can be continued analytically to all complex values s ≠ 1. For s = 1 the series is the harmonic series which diverges to +∞, and

 \lim_{s\to 1}(s-1)\zeta(s)=1.

Thus the Riemann zeta function is a meromorphic function on the whole complex s-plane, which is holomorphic everywhere except for a simple pole at s = 1 with residue 1.  (A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain.) (A meromorphic function is a ratio of two well-behaved (holomorphic) functions.)

The values of the Riemann zeta function at even positive integers were computed by Euler. The first of them, ζ(2), provides a solution to the Basel problem.   In 1979 Apéry proved the irrationality of ζ(3). The values at negative integer points, also found by Euler, are rational numbers.

Riemann zeta function ζ(s) in the complex plane. The color of a point s encodes the value of ζ(s): colors close to black denote values close to zero, while hue encodes the value's argument.

The white spot at s = 1 is the pole of the zeta function; the black spots on the negative real axis and on the critical line Re(s) = 1/2 are its zeros. Values with arguments close to zero including positive reals on the real half-line are presented in red.

    
This image shows a plot of the Riemann zeta function along the critical line for real values of t running from 0 to 34. The first five zeros in the critical strip are clearly visible as the place where the spirals pass through the origin.

Leonhard Euler is most famous for the "Euler Identity":  e^{ix} = \cos x + i\sin x \                                                           The special case, with x = π  gives the beautiful identity:  e^{i \pi} + 1 = 0, which involves 0, 1, i, e and π. 

We can easily see this result by looking at the diagram of a circle. 

The x axis contains the real numbers (Re) and the y axis contains the imaginary numbers (Im). 

The radius of the circle is i

The identity is when the angle equals  π

Divergent Series

Under many circumstances, it is desirable to assign a limit to a series which fails to converge in the usual sense. A summability method is such an assignment of a limit to a subset of the set of divergent series which properly extends the classical notion of convergence.

In the fourteenth century mathematicians studied the Harmonic Series: the sum of the reciprocals of the natural numbers.

\sum_{n=1}^\infty\,\frac{1}{n} \;\;=\;\; 1 \,+\, \frac{1}{2} \,+\, \frac{1}{3} \,+\, \frac{1}{4} \,+\, \frac{1}{5} \,+\, \cdots.\!

Its name derives from the concept of overtones, or harmonics in music: the wavelengths of the overtones of a vibrating string are 1/2, 1/3, 1/4, etc., of the string's fundamental wavelength. Every term of the series after the first is the harmonic mean of the neighboring terms; the phrase harmonic mean likewise derives from music.

This series diverges slowly. For the partial sums ps we see:

n 6 100 1,000 12,367 272,400,600 1.5 x 10 to the 43rd*
ps 2.45 5.187... 7.486... 10... 20... 100....
* Calculated exactly by John W Wrench Jr. in 1968.   See http://www.maa.org/sites/default/files/269058023489.pdf  and also http://books.google.com/books?id=7sDtIy8MNIC&pg=PA23&lpg=PA23&dq=john+wrench+in+1968+harmonic+series&source=bl&ots=
9MQDjOo8Ya&sig=_Ew4FuVr8H_dY7DNeBhU31NjlU8&hl=en&sa=X&ei=IatTU4_ZMsjIsASNwIAw&ved=0CC0Q6AEwAQ#v=onepage&q=j
ohn%20wrench%20in%201968%20harmonic%20series&f=false
 
The partial sums also avoid all integers!  The partial sums must always be a fraction, finite or infinite.  For n = 1, 2, and 6 they are 1, 1.5, and 4.45 and infinitely recurring for all other n.

The general harmonic series is of the form

\sum_{n=0}^{\infty}\frac{1}{an+b} ,\!

where a \ne 0 and b are real numbers. The general harmonic series diverge.

The sum of harmonic progression is mathematically related to derivative of gamma function

   1 \,+\, \frac{1}{2} \,+\, \frac{1}{3} \,+\, \frac{1}{4} \,+\, \frac{1}{5} \,+\, \cdots\, \frac{1}{n}   =  \left(\frac{{\rm d}}{{\rm d}x}\,\ln(\Gamma(x+1))\right)_{x=n} +\, \gamma.

Here 'x' is the number of terms up to which sum is taken.  \gamma is Euler–Mascheroni constant: 0.57721566...

Another slowly divergent series is the sum of the reciprocals of all prime numbers:
 
\sum_{p\text{ prime }}\frac1p = \frac12 + \frac13 + \frac15 + \frac17 + \frac1{11} + \frac1{13} + \frac1{17} + \cdots = \infty

This was proved by Leonhard Euler in 1737, and strengthens Euclid's 3rd-century-BC result that there are infinitely many prime numbers.

There are a variety of proofs of Euler's result, including a lower bound for the partial sums stating that

\sum_{\scriptstyle p\text{ prime }\atop \scriptstyle p\le n}\frac1p \ge \log \log (n+1) - \log\frac{\pi^2}6

for all natural numbers n. The double natural logarithm indicates that the divergence might be very slow, which is indeed the case, see Meissel–Mertens constant. The partial sum for the primes less than 1 million is only 2.887289... .

The Meissel–Mertens constant (named after Ernst Meissel and Franz Mertens) defined as the limiting difference between the harmonic series summed only over the primes and the natural logarithm of the natural logarithm:

M = \lim_{n \rightarrow \infty } \left(
\sum_{p \leq n} \frac{1}{p}  - \ln(\ln(n)) \right)=\gamma + \sum_{p} \left[ \ln\! \left( 1 - \frac{1}{p} \right) + \frac{1}{p} \right].

The symbol γ is the famous Euler–Mascheroni constant, which has a similar definition involving a sum over all integers (not just the primes).  The value of M is approximately  0.2614972128.... .

Trivia: The number M was used by Google when bidding in the Nortel patent auction. Google posted three bids based on mathematical numbers: $1,902,160,540 (Brun's constant), $2,614,972,128 (Meissel–Mertens constant), and $3.14159 billion ( π ). Google  famously bid pi billion dollars and other multiples of mathematical constants, showing a strange sense of humor with shareholders' money.

An example of mathematical elegance is Leibnitz's series:

\sum_{n=0}^\infty \, \frac{(-1)^n}{2n+1} \;=\; \frac{\pi}{4}.\! 1 \,-\, \frac{1}{3} \,+\, \frac{1}{5} \,-\, \frac{1}{7} \,+\, \frac{1}{9} \,-\, \cdots \;=\; \frac{\pi}{4}.\!

GAMMA FUNCTION

The (complete) gamma function Gamma(n) is defined to be an extension of the factorial to complex and real number arguments. It is related to the factorial by  Gamma(n)=(n-1)!, a slightly unfortunate notation due to Legendre which is now universally used instead of Gauss's simpler Pi(n)=n!  There are no points z at which Gamma(z)=0.

The gamma function can be defined as a definite integral for R[z]>0(Euler's integral form) as

  Gamma(z)=int_0^1[ln(1/t)]^(z-1)dt. or int_0^inftyt^(z-1)e^(-t)dtor 2int_0^inftye^(-t^2)t^(2z-1)dt,

The complete gamma function Gamma(x) can be generalized to the upper incomplete gamma function Gamma(a,x)and lower incomplete gamma function gamma(a,x).

GammaReImAbs

Plots of the real and imaginary parts of Gamma(z)in the complex plane are illustrated above.

Below we see the gamma function along part of the real axis: 

GammaFunction

Integrating   Gamma(z)2int_0^inftye^(-t^2)t^(2z-1)dt, by parts for a real argument, it can be seen that Gamma(x)(x-1)Gamma(x-1).

If  x is an integer  Gamma(n) = (n-1)!, so the gamma function reduces to the factorial for a positive integer argument.

A beautiful relationship between Gamma(z)and the Riemann zeta function zeta(z)is given by:

 zeta(z)Gamma(z)=int_0^infty(u^(z-1))/(e^u-1)du   for R[z]>1

The gamma function can also be defined by an infinite product form:

 Gamma(z)=[ze^(gammaz)product_(r=1)^infty(1+z/r)e^(-z/r)]^(-1),  where gammais the Euler-Mascheroni constant

The Euler limit form is  Gamma(z)=1/zproduct_(n=1)^infty[(1+1/n)^z(1+z/n)^(-1)], so, Gamma(z) = lim_(n->infty)(n!)/((z)_(n+1))n^z

The reciprocal of  the gamma function 1/Gamma(z) is an entire function  expressed as

  1/(Gamma(z))=zexp[gammaz-sum_(k=2)^infty((-1)^kzeta(k)z^k)/k],  

where gammais the Euler-Mascheroni constant and zeta(z)is the Riemann zeta function 

 

 

 

 

 

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