https://www.youtube.com/watch?v=7y3gHO-Pz2o An incredible fact of Euclidean geometry is that the exterior angles of any convex polygon sum to $360^{circ}$, which is independent of the number of sides of the polygon. We prove this fact here using the formula for the sum of the interior angles of a convex $n$-gon.
Exterior Angle Theorem Read More »
A basic yet amazing fact of Euclidean geometry is that the sum of the interior angles of a convex polygon with $n$ sides is equal to $$180^{circ} (n-2),$$ which is a formula that depends only on $n$. We show how to prove this result by dissecting the polygon into triangles.
Interior Angle Theorem Read More »
https://www.youtube.com/watch?v=z25imPmrxR0 Many people have seen formulas for the sum of the first $n$ positive integers, or the sum of their squares or cubes. But what about finding a formula for the sum of the $n^{text{th}}$ powers of these integers? We show a recursive method of finding polynomial formulas for $$1^k + 2^k + cdots +
Sum of Powers of Positive Integers Read More »
https://www.youtube.com/watch?v=IYZQXV3SA0I Two important techniques in the ad hoc evaluation of series are telescoping and partial fraction decomposition. The latter is useful in calculus too, as an integration technique for rational functions. We show an example here of an infinite series $$frac{1}{1cdot 2 cdot 3}+frac{1}{2cdot 3cdot 4}+cdots$$ that uses both telescoping and partial fraction decomposition in
Telescoping & Partial Fraction Decomposition Read More »
https://www.youtube.com/watch?v=wqHFq1OqmD4 Where an arithmetic series adds a common difference to each successive term, a geometric series multiplies successive terms by a common ratio. We show how to evaluate such a series, both in its finite and infinite variants. The infinite variant is one of the simplest infinite series that actually converge, and its formula is
https://www.youtube.com/watch?v=qrYlQc0j1Ao At a young age, Gauss shocked his teacher by finding the sum of the first $100$ positive integers quickly using the formula $$1+2+cdots+n=frac{n(n+1)}{2}.$$ We use a generalization of his techique to find the sum of an arithmetic series.
Arithmetic Series Using Gauss’s Formula Read More »
https://www.youtube.com/watch?v=EEI6ucn5FS8 We show the analogue of the triangle inequality when floor and ceiling functions replace absolute value. This results in some interesting inequalities that are useful in obtaining approximation results. The floor function one is $$0le lfloor x+y rfloor -lfloor x rfloor – lfloor y rfloor le 1.$$
Floor and Ceiling Sum Bounds Read More »
https://www.youtube.com/watch?v=sBD3fp0XLT0 Hermite’s identity is a remarkable identity that allows us to simplify certain sums where each term involves the floor function. It comes up in math contests and olympiads from time to time. We prove this identity using an unexpected technique, namely periodicity.
Hermite’s Floor Identity Read More »
https://www.youtube.com/watch?v=PEQ1-tjlnmo The reverse triangle inequality $$|x-y|ge ||x|-|y||$$ is a classic textbook exercise using the ordinary triangle inequality, but it is also a useful result in calculus or mathematical analysis. We prove it here using two different methods.
Reverse Triangle Inequality Read More »
https://www.youtube.com/watch?v=mUhBemtfivQ The triangle inequality $$|x|+|y|ge |x+y|$$ for real numbers is an indispensable tool in mathematical analysis, otherwise known as calculus, because it allows us to approximate quantities by bounds. We prove a general version of it using the ordinary absolute value and for any number of variables.
Real Triangle Inequality Read More »