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
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k2
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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.
Niklov Rother (Johns Hopkins University) Euler Approximation for ∑∞k=1
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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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k2
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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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k2
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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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k2
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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
Niklov Rother (Johns Hopkins University) Euler Approximation for ∑∞k=1
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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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k2
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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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k2
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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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k2
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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.
Niklov Rother (Johns Hopkins University) Euler Approximation for ∑∞k=1
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k2
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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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k2
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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.
Niklov Rother (Johns Hopkins University) Euler Approximation for ∑∞k=1
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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
