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 examples of Bernoulli’s equation, see “What moves fast, will slow down” and “An Epilogue to ‘A Relentless Pursuit’“.
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.