The differential equation
where and , is known as the Bernoulli’s equation.
When is an integer, (1) has trivial solution .
To obtain nontrivial solution, we divide each term of (1) by to get,
(2) can be expressed as
Let , (3) is transformed to a first order linear equation
giving the general solution of a Bernoulli’s equation (see Fig. 1)
For a concrete example of Bernoulli’s equation, see “What moves fast, will slow down“
Summations arise regularly in mathematical analysis. For example,
Having a simple closed form expression such as makes the summation easier to understand and evaluate.
The summation we focus on in this post is
We will find a closed form for it.
In a recent post, I derived the closed form of a simpler summation (see “Beer theorems and their proofs“) Namely,
From (2) it follows that
which gives us
Let we arrived at (1)’s closed form:
I have a Computer Algebra aided solution too.
Therefore, the closed form of is the solution of initial-value problem
It is solved by Omega CAS Explorer (see Fig. 1)
At ACA 2017 in Jerusalem, I gave a talk on “Generating Power Summation Formulas using a Computer Algebra System“.
I had a dream that night. In the dream, I was taking a test.
Derive the closed form for
I woke up with a sweat.
To prove Beer Theorem 2 (see “Beer theorems and their proofs“) is to show that the Harmonic Series diverges.
Below is my shot at it.
Yaser S. Abu-Mostafa proved a theorem in an article titled “A differentiation test for absolute convergence” (see Mathematics Magazine 57(4), 228-231)
His theorem states that
Let be a real function such that exists at . Then converges absolutely if and only if .
Let , we have
the Harmonic Series. And,
Therefore, by Abu-Mostafa’s theorem, the Harmonic Series diverges.