Most people are familiar with the Pythagorean theorem: In a right-angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides. As the name of the theorem implies, it is attributed to Pythagoras, a Greek mathematician who lived around 500 B.C. The theorem is also included [...]
Archives for cool maths
Remembering Trigonometric Addition Formulas
The addition formulas for sine and cosine look like this:
\begin{aligned}
\cos(\alpha + \beta) &= \cos \alpha \cos \beta – \sin \alpha \sin \beta, \\
\sin(\alpha + \beta) &= \sin \alpha \cos \beta + \cos \alpha \sin \beta. \\
\end{aligned}
I can never remember them.
On the Divergence of a Geometric Progression Sum
Let us revisit the geometric progression sum considered in an earlier article,
s_r = \sum_{k=0}^\infty r^k = 1 + r + r^2 + r^3 + \ldots,
where r here is a complex number. For what values of r does this infinite sum make sense? Can we find a closed-form expression for s_r in such cases? To investigate [...]
Nice Proof of a Geometric Progression Sum
Consider the geometric series,
s_r = \sum_{k=0}^\infty r^k = 1 + r + r^2 + r^3 + \ldots,
for 0 < r < 1. The goal is to find a closed-form expression for s_r.
