1647 words - 7 pages

Euler Approximation for

∞

∑

k=1

1

k2

Niklov Rother

Johns Hopkins University

December 16, 2018

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1

k2

December 16, 2018 1 / 25

What will be covered

1 History and Background

Biography Of Euler

Prerequisites

2 Proof

Proof 1

Proof 2

3 Conclusion

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Biography

Leonhard Euler, born April 15, 1707, Basel, Switzerland—died

September 18, 1783, St. Petersburg, Russia

Swiss mathematician and physicist, one of the founders of pure

mathematics.

He made decisive and formative contributions to the subjects of

geometry, calculus, mechanics, and number theory

He also developed methods for solving problems in observational

astronomy and demonstrated useful applications of mathematics in

technology and public affairs.

Much of the notation used by mathematicians today - including

e, i , f (x),∑, and the use of a, b and c as constants and x , y and z as

unknowns - was either created, popularized or standardized by Euler.

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Biography

Euler’s Formula eθi = cos(θ) + isin(θ)

He produced one of the most beautiful, mathematical equations,

e ipi = −1

1735, Euler solved an intransigent mathematical and logical problem,

known as the Seven Bridges of Ko¨nigsberg Problem

The demonstration of geometrical properties such as Euler’s Line and

Euler’s Circle;

A new method for solving quartic equations;

The Prime Number Theorem, which describes the asymptotic

distribution of the prime numbers;

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Weierstrass Factorization Theorem

Firstly, any finite sequence {cn} in the complex plane has an

associated polynomial p(z) that has zeroes precisely at the points of

that sequence,

p(z) = ∏

n

(z − cn).

Secondly, any polynomial function p(z) in the complex plane has a

factorization

p(z) = a∏

n

(z − cn)

where a is a non-zero constant and {cn} are the zeroes of p.

It is a necessary condition for convergence of the infinite product in

question is that for each z, the factors (z − cn) must approach 1 as

n→ ∞ .

∏bi=a f (i) This is simply the multiplication of all values from f (a) to

f (b)

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Riemann Zeta

The Riemann zeta function ζ(s) is one of the most significant

functions in mathematics because of its relationship to the

distribution of the prime numbers.

The zeta function is defined for any complex number s with real part

greater than 1 by the following formula:

ζ(s) =

∞

∑

n=1

1

ns

This is where the infinite summation we are dealing with, come from.

ζ(2) =

∞

∑

n=1

1

n2

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Sine Function

f (x) = sin(x) is a very well known trigonometric function

This infinite collection of x-values at which sin x equals zero reflects the

repeating, periodic behavior of the sine function.

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Sine Function

Sine function is a periodic function, meaning that it is a function that

repeats its values in regular intervals or periods.

We note that that sin(x) has an infinite amount of roots or zeros when

x = 0, x = ±pi,±2pi,±3pi.....

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Sine Function- Taylor series

Calculus: Taylor series is a way to approximate the value of a function by

taking the sum of its derivatives at a given point. It is a series expansion

around a point.

Note: the expression for sin x will continue forever, with the powers on the

x running through the sequence of odd integers, the denominators being

the associated factorials, and the signs alternating between positive and

negative.

Note: Factorials: 3! = 3 · 2 · 1 = 6

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Taylor series and Infinite Polynomials

Suppose P(x) is a polynomial of degree n having as its n roots

x = a, x = b, x = c, ..., and x = d ;

Because they are roots

P(a) = P(b) = P(c) = ... = P(d) = 0.

Initially Euler started out by saying

P(x) = (x − a)(x − b)(x − c)...(x − d)

However this polynomial tends toward infinity. We require it to

converge to a non-zero limit. Euler modified the entire polynomial by

dividing each linear term by the corresponding root, and the product

of these on the outside of the polynomial as a constant.

We also let P(0) = 1. Euler knew that P(x) factors into the product

of n linear terms as follows:

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Taylor series and Infinite Polynomials

We make a generalized note that P(a) = 0

We can also note that for each root we substitute in, the specific

linear term that contains that root, becomes zero, in turn making the

entire polynomial zero.

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Taylor series and Infinite Polynomials

We also want the polynomial to follow the condition that P(0) = 1.

We test this by substituting x for 1, as follows,

We now have the infinitely long polynomial P(x), with the condition

that it has roots at x = a, x = b, x = c , ...x = d , or rather

P(a) = P(b) = P(c) = ... = P(d) = 0, and that P(0) = 1

Euler uses this definition of an infinitely long polynomial in his

Approximation of the value of pi

2

6

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Eulers Proof: ζ(2)

1 Let f (x) be an infinite polynomial, f (0) = 1

2 We then decide to factor out an x as follows,

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Eulers Proof: ζ(2)

1 This then reduces to

We note here that this initial series is the Taylor expansion of sin(x)x

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Eulers Proof: ζ(2)

1 We then find the zeros, or roots of the function.

sin(x)

x = 0

Note: x 6= 0

We then get that sin(x) = 0

this is zero when x = 0, x = ±pi,±2pi,±3pi.....

Now we only look at this when

x = ±pi,±2pi,±3pi.....

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Eulers Proof: ζ(2)

1 Because these Numbers are all zeros, we can factor them into an

infinite series of binomials.

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Eulers Proof: ζ(2)

1 We then combine the similar binomials together to form

We note that we have turned an infinite summation into an infinite product

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Approximation of pi

1 We now Take this infinite product and begin multiplying out the terms

2 the first term of this newly expanded product of all of the terms

would be 1.

3 We end up with the second term being a factor of x2 This comes

from the the the multiplication of all of the 1’s with all but one of the

factors by an x2 term from that remaining factor.

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Approximation of pi

1 Now we have 2 infinite sums, as seen below

2 Note that both series begin with 1, and the series both alternate,

identically from positive to negative to positive and so on.

3 We can now equate the two x2 terms from both infinite sums.

4 We then Multiply both sides by −1 to yield

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Approximation of pi

1 We then cross-multiply and get that

Q.E.D

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Using Cosine

Here we start with the cosine function instead

cos(x) can also be expressed by the following Maclaurin series

expansion:

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Using Cosine

Comparing the x2 coefficients gives:

Thus

We note that this is the sum of all the reciprocals of the odd numbers.

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Using Cosine

Note that

This then would suggest that

And

Niklov Rother (Johns Hopkins University) Euler Approximation for ∑∞k=1

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Using Cosine

Taking from what we know,

∞

∑

n=1

1

(2n− 1)2 =

∞

∑

k is odd

1

k2

=

pi2

8

=

3

4

ζ(2)

Solving for the ζ(2) we get,

ζ(2) = pi

2

6

Q.E.D

Niklov Rother (Johns Hopkins University) Euler Approximation for ∑∞k=1

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k2

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Conclusion

Beautiful and intuitive proof

Easy to follow conceptually

Niklov Rother (Johns Hopkins University) Euler Approximation for ∑∞k=1

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k2

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History and Background

Biography Of Euler

Prerequisites

Proof

Proof 1

Proof 2

Conclusion

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