1647 words - 7 pages

Euler Approximation for

∞

∑

k=1

1

k2

Niklov Rother

Johns Hopkins University

December 16, 2018

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

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

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

1

k2

December 16, 2018 2 / 25

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.

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

1

k2

December 16, 2018 3 / 25

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;

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

1

k2

December 16, 2018 4 / 25

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)

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

1

k2

December 16, 2018 5 / 25

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

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

1

k2

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Sine Function

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

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