In this post, I wish to find the value of some finite product. I begin by defining a function as follows and trying to find it’s roots
Let , then we have
and similarly . So we have
The sine function vanishes at integer multiples of , so it follows that where for all integers , that is, for for all . Thus, for
where we recall that . Since is strictly increasing on the interval , it follows that
moreover, since tangent is an odd function, we have for each . In particular we have found distinct roots of , so as a consequence of the fundamental theorem of algebra, we can write
So we have , and
Our goal in this post is to provide an elementary proof for the real version of Euler’s famous Sine expansion.
I first prove that for any that is a power of and for any ,
by using the following more familiar identity
we can easily arrive to the following identity that is valid for equal to any power of ,
By considering what happens as in the recent formula, we can obtain that for a power of
and then I replace in and prove .
Now I choose and break up as follows
Since for , , we have
and if I choose and large enough such that , then
which implies that
Therefore from we have
Also we have
and once again by , we get
To summarize, I have shown that
Now taking in I arrive to
After rearrangement and then taking absolute values, we get
but since we have
I can write
Finally since , this inequality implies our goal, indeed
In this post I want to find the value of the following finite sum,
I use the following amazing identity that you can see it’s proof in my previous posts
By this identity, I can write
and this follows that
Sums of the form where , sometimes are called Euler sums. There are several ways to evaluate these sums. Aabout 240 years ago, Leonhard Euler(1707-1783) in 1775 proved that for ,
In the following you can see an elementary proof of this formula in the three steps:
Now by recent formula that is also an alternate definition of the Harmonic numbers, we can write:
By using the Flajolet-Vardi theorem we can find the value of the another amazing convergent series. Indeed, following series
I first recall the Flajolet-Vardi theorem that you can find it’s proof in my second post:
If and converges then,
This theorem shows that , because if we let , then and by this theorem
Therefore now we must find the value of . we use the Taylor expansion of and the fact that the sum is if divides and otherwise.
There are some ‘s of complex numbers. Those numbers have always non-negative real part, for the Argument we take the angle between and , so that it fits with the power series for .
We thus have
We have to take the limit . The term disappears, so we get
For we have
and also for ,
In this post I want to generalize the given formula at my second post as follows
To prove this , I first prove following useful identity
PROOF: (Trigonometric Method) Note that
Repeatedly applying , we arrive at the following formula:
and now note that
By using this recent identity we can write
One another way that can evaluate by double integral is to write as follows
This is true, since we have
Now integrating by parts yields
and by using the monotone convergence theorem, I can write
The Italian mathematician Pietro Mengoli (1625–1686), in his 1650 book Novae quadraturae arithmeticae, seu de additione fractionum, posed the following question: What’s the value of the sum
Here’s what Mengoli said:
Having concluded with satisfaction my consideration of those arrangements of fractions,
I shall move on to those other arrangements that have the unit as numerator, and square
numbers as denominators. The work devoted to this consideration has bore some fruit — the question itself still awaiting solution — but it [the work] requires the support of a richer
mind, in order to lead to the evaluation of the precise sum of the arrangement [of fractions]
that I have set myself as a task.
First time, Leonhard Euler (1707-1783) in 1735 proved that above series converges to .
In page 122 of the book “Topics in Number Theory”(1956), by William J. LeVeque there is an exercise for evaluating the following integral in two ways.
First way is to write the integrand as a geometric series,
and the second way by use of a suitable change of variables () which is also published by Tom M. Apostol in 1983 in Mathematical Intelligencer.
and by the monotone convergence theorem we get
We change variables in this by putting , so that . Hence
where is the square with vertices and .
Exploiting the symmetry of the square we get
Now , and if then and .
It follows that and so . Hence
In this post I want to find the value of the sum
Note that for , .
I use following formula that is called Gregory-Leibniz-Madhava’s series
If we define function as follows:
Now I use this theorem
Hence, by Cauchy’s double series theorem, we can switch the order of summation:
This theorem implies that
My first post is about two properties in the elementary set theory. You can see an amazing proof for them from me in the following.
In proof I used the following property in logic
that is called “constructive dilemma”.
Let , then I can write
which implies that .
Let , then
which implies that .