Mathematical Odds & Ends

In this recent post, John Cook stated that the product of 4 consecutive Fibonacci numbers is equal to the product of two consecutive integers.

For any natural number , let denote the sum of its factors including itself.

In this previous post, I claimed that , where is the zeta function There are several proofs for .

The zeta function is defined as In Reference 1, I learnt of a pretty simple double integral which evaluates to , which in turn is well-known to be equal to .

The zeta function is defined as and the gamma function is defined as The integral representation of the zeta function refers to the following relationship: Proposition.

Nahin’s In Pursuit of Zeta-3 (Reference 1) is full of many interesting theorems and results connecting different important functions in math.

The Gaussian integral, sometimes known as the probability integral, is the result In In Pursuit of Zeta-3 (Reference 1), Nahin presents and proves a generalization of the Gaussian integral: for any fixed constants , (We obtain the Gaussian integral by set…

I learnt of the following statement from John Cook’s blog: Let and denote the golden ratio and its inverse respectively: If a function is such that , , is strictly non-decreasing and differentiable, and for all , Then we must have for all .

Evaluate Solution: For any finite , The first product is easy to reduce: If you write our the first few terms of the second product, it’s not hard to come to the following in order to get telescoping cancellation: Putting the two together,

The napkin ring problem concerns finding the volume of a solid after a hole has been drilled into a sphere: Take a sphere of radius r and drill out a hole along a diameter so the remaining shape has height h.

The Ford circles, introduced by L.