Math

Sums

Each sum of the form ${\sum }_{x=1}^n{x}^k=1^k+2^k+\cdots +n^k$ has a closed-form formula that is a polynomial of degree k+1 ${\sum }_{x=1}^n{x}^2=1^2+2^2+3^2+\cdots n^2=\frac{n\left(n+1\right)\left(2n+1\right)}{6}$. There is a general form for this but its complex.

An arithmetic progression is a sequence of numbers where the difference between and two numbers is constant. This can be represented as $a+\cdots +b=\frac{n\left(a+b\right)}{2}$ for n numbers.

A geometric progression is a sequence of numbers where the ratio between two consecutive numbers is constant (ex. $n_2/n_1=3$). The sum of a geometric progression can be computed as $a+ak+a{k}^2+\cdots +b=\frac{bk-a}{k-1}$.

Where a is the first number and b is the last number and k is our ratio.

A harmonic sum is a sum of the form ${\sum }_{x=1}^n\frac{1}{x}=1+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{n}$ an upper bound for a harmonic sum is ${\log }_2\left(n\right)+1$. We can modify each $\frac{1}{k}$ term so $k$ becomes the nearest power of two that does not exceed k. For example… $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}\le 1+\frac{1}{2}+\frac{1}{2}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}$

???

I have no idea what utility this provides since these are not equivalent terms. I’m on a plane right now so I cant look this up lol.