Reversal of Indices in Infinite Triangular Sums

Friday, 31 January 2014

We have:

\sum_{i=0}^{\infty} \sum_{j=i}^{\infty} \varphi(i,j) = \sum_{j=0}^{\infty} \sum_{i=0}^{j} \varphi(i,j)


\sum_{i=0}^{\infty} \sum_{j=i}^{\infty} \varphi(i,j) = \sum_{\begin{array}{c}i,j \\ 0 \le i \le j < \infty\end{array}} \varphi(i,j) = \sum_{j=0}^{\infty} \sum_{i=0}^{j} \varphi(i,j)

In an earlier post, I showed the finite version of this result:

\sum_{i=0}^{n} \sum_{j=i}^{n} \varphi(i,j) = \sum_{j=0}^{n} \sum_{i=0}^{j} \varphi(i,j)

This might be seen to hold as both sums are

\sum_{\begin{array}{c}i,j \\ 0 \le i \le j \le n\end{array}} \varphi(i,j)


Fibonacci Formulae

Thursday, 21 November 2013

Whilst doodling with the Fibonacci sequence

n 0 1 2 3 4 5 6 7 8 9 10
Fn 0 1 1 2 3 5 8 13 21 34 55

I found some interesting formulae:

  • F_n = \displaystyle\sum_{k=0}^{\left\lfloor\frac{n-1}{2}\right\rfloor} \binom{n-1-k}{k}
  • F_{2m} = - \dfrac{1}{2} \displaystyle\sum_{k=0}^{2m-1} (-1)^k \binom{2m}{k} F_k
  • \displaystyle\sum_{k=0}^{2m} (-1)^k \binom{2m+1}{k} F_k = 0

The first of these is not new, but I did not find the other two on the web.

There’s more information about the Fibonacci sequence and the binomial coefficients on Wikipedia.


An Intuitive Representation of the Eisenstein Integers

Wednesday, 6 November 2013

The Eisenstein integers

{\mathbb Z} [ \omega ]

where ω is a primitive cube root of 1 given by

\omega = \dfrac{-1 + {\mathrm {i}} \sqrt{3}}{2} = \sqrt[3]{1}

are often represented in the form

\{ a + b \cdot \omega \;|\; a, b \in {\mathbb Z}\}
(noting that \omega^2 = - (1 + \omega))

and perhaps abbreviated by pair notation such as

\{ [ a ; b ]^{*} \;|\; a, b \in {\mathbb Z}\}

(where the asterisk is to distinguish this notation from that introduced below)

so that

[ a ; b ]^{*} + [ c ; d ]^{*} = [a+b \;;\; c+d]^{*}

[ a ; b ]^{*} \times [ c ; d ]^{*} = [(a c - b d) \;;\; (b c + a d - b d) ]^{*}

Here is a more intuitive representation that is simpler to manipulate and reason about.

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Sublime Text Tip: Indicating Current and Edited Tabs

Friday, 25 October 2013

To colour the current tab green, and any tab of an edited file red, add the following to your theme file.

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Hyperbolic Mnemonic

Friday, 11 October 2013

If you have problems remembering which is which with sinh and cosh, or their graphs, or which way around their definitions are:

\sinh x = \dfrac{\mathrm{e}^x - \mathrm{e}^{-x}}{2}

\cosh x = \dfrac{\mathrm{e}^x + \mathrm{e}^{-x}}{2}

then it might help if you notice that the graph of sinh is somewhat S-shaped (though backwards), and cosh is C-shaped (though open upwards).

[Well, unless you’re Russian, where the Cyrillic letter Es (‘С’) corresponds to our ‘S’; that might throw a graphemic spanner in the works.]

Derivation Summary

Friday, 11 October 2013

Here is a summary of rules of derivation (sometimes referred to as differentiation).

I wrote this post as the notation used on other sites is often ambiguous (particularly with juxtapositions that could denote either a product or a function application).

I previously wrote some brief notes on some derivations of the derivation formulae.

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Surface Areas and Volumes of Hyperspheres

Thursday, 3 October 2013

I noticed that the formulae for the sizes of the surface (or rather, boundary) and volume (interior) of circles, spheres and hyperspheres seem to have discontinuities (or ‘jumps’) in the index of \pi.

However, this is only a result of simplifying a uniform formula for particular cases.

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On Fibonacci-style Sequences continued

Tuesday, 1 October 2013

In my previous post on Fibonacci-style sequences, I gave the formulae for the closed form

a_n = p \cdot r^n + q \cdot s^n

given the recurrence values a_0, a_1, b, c where

a_{n+2} = b \cdot a_{n} + c \cdot a_{n+1}

but I did not show that the formula is valid. I do that here.

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On Fibonacci-style Sequences

Sunday, 29 September 2013

The Fibonacci sequence F_0, F_1, F_2, \dots which begins 0, 1, 1, 2, 3, 5, \dots is defined by

F_{0} = 0

F_{1} = 1

F_{n+2} = F_{n} + F_{n+1}

(the starting values vary by author).

This is a special case of a more general sequence given by



a_{n+2} = b \cdot a_{n} + c \cdot a_{n+1}

There are at least three ways to determine such a sequence. I explore those here.

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Books – Non-fiction – Text Books

Sunday, 29 September 2013

Serious, technical books I recommend.

These are listed in no particular order

The Continuity of Life
Daniel J. Fairbanks, W. Ralph Andersen

Lectures on Physics
Richard Feynman

Concrete Mathematics
Ronald L. Graham, Donald E. Knuth and Oren Patashnik