Daily math problems for June 23, 2026 — Functions, Matrices, Sequences and series

We need $x-2\ge 0$, i.e. $x\ge 2$. The domain is $[2,+\mathrm{\infty })$.

$\bullet$ Evaluate each of the nine limits to fill the matrix:

$$\left(\begin{array}{ccc}\sqrt{3}\mathrm{/}2& 2& 1\\ 3& 4& 4\\ 0& 3& 1\end{array}\right)$$

$\bullet$ Expand along row 1. The cofactor of the $(1,1)$ entry is $4\cdot 1-4\cdot 3=-8$:

$$\frac{\sqrt{3}}{2}\cdot (-8)=-4\sqrt{3}$$

$\bullet$ Cofactors of the $(1,2)$ and $(1,3)$ entries are $3\cdot 1-4\cdot 0=3$ and $3\cdot 3-4\cdot 0=9$:

$$-2\cdot 3+1\cdot 9=3$$

$\bullet$ Add the three contributions:

$$\det =3-4\sqrt{3}$$

Euler's classical result: ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}{\displaystyle \frac{1}{n^2}}={\displaystyle \frac{{\pi }^2}{6}}}$.