PEEWSRIGWETRYqx

2022-01-01

To me the most common identity that comes to mind that results in 1 is the trigonometric sum of squared cosine and sine of an angle:
${\mathrm{cos}}^{2}\theta +{\mathrm{sin}}^{\theta }=1$ (1)
and maybe
$-{e}^{i\pi }=1$ (2)
Are there other famous (as in commonly used) identities that yield 1 in particular?

jgardner33v4

Expert

Pick your favorite probability density function f, then
${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f\left(x\right)dx=1$

Tiefdruckot

Expert

$\left(\mathrm{\forall }x\in \mathbb{R}\right){\mathrm{cos}h}^{2}\left(x\right)-{\mathrm{sin}h}^{2}\left(x\right)=1$

Vasquez

Expert

Here are two well-known infinite series whose sum is 1:
$\sum _{n=1}^{\mathrm{\infty }}\frac{1}{{2}^{n}}=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\dots =1$
$\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n\left(n+1\right)}=\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\frac{1}{4\cdot 5}+\frac{1}{5\cdot 6}+\dots =1$
The following series is less known, interesting although:
$\sum _{n=1}^{\mathrm{\infty }}\frac{1}{{s}_{n}}=\frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{43}+\frac{1}{1807}+\dots =1$
where denominators form Sylvester sequence: every term is equal to the product of all previous terms, plus one. For example
2
3=2+1
7=2 * 3+1
43=2 * 3 * 7+1
1807=2 * 3 * 7 * 43+1
and so on

Do you have a similar question?