Cyclic sum

A cyclic sum is a summation that cycles through all the values of a function and takes their sum.

Rigorous definition
Consider a function $f(a_1,a_2,a_3,\cdots,a_n)$. The cyclic sum $\sum\limits_{\text{cyc}}f(a_1,a_2,a_3,\cdots,a_n)$ is equal to $$f(a_1,a_2,a_3,\cdots,a_n)+f(a_2,a_3,a_4,\cdots,a_n,a_1)+\cdots+f(a_n,a_1,a_2,\cdots,a_{n-2},a_{n-1}).$$ The notation $\sum\limits_{\text{cyc}}$ implies that all variables are cycled through. Another notation is $\sum\limits_{a,b,c}$, which implies that the cyclic sum only cycle through those variables underneath the sigma. [Note: Do not confuse this notation with the symmetric sum.]

Examples
Consider the permutation $p=(a\;b\;c)$. The cyclic sum $\sum\limits_p a$ is the sum that cycles through the permutation: $$\sum_p a=a+b+c.$$
They often come up in inequalities: $$\sum_{a,b,c}\frac{a^3}{3}=\frac{a^3}{3}+\frac{b^3}{3}+\frac{c^3}{3}\geq \sqrt[3]{\frac{(abc)^3}{3^3}}=\frac{abc}{3}.$$
They are extremely helpful in inequalities involving many letters. Instead of writing all the terms of the sum explicitly, we can employ this notation. Check out this answer.

Cyclic numbers
$142857$, the six repeating digits of $\frac{1}{7}$, $0.\overline{142857}$, is the best-known cyclic number in base $10$.
$$1\times 142,857=142,857\\ 2\times 142,857=285,714\\ 3\times 142,857=428,571\\ 4\times 142,857=571,428\\ 5\times 142,857=714,285\\ 6\times 142,857=857,142\\ 7\times 142,857=999,999$$
When multiplied by $2, 3, 4, 5$, or $6$, the answer will be a cyclic permutation of itself, and will correspond to the repeating digits of $\dfrac{2}{7},\dfrac{3}{7},\dfrac{4}{7},\dfrac{5}{7},\dfrac{6}{7}$, respectively.
$$1\div 7=0.\overline{142,857}\\ 2\div 7=0.\overline{285,714}\\ 3\div 7=0.\overline{428,571}\\ 4\div 7=0.\overline{571,428}\\ 5\div 7=0.\overline{714,285}\\ 6\div 7=0.\overline{857,142}\\ 7\div 7=0.\overline{999,999}=1\\ 8\div 7=1.\overline{142,857}\\ 9\div 7=1.\overline{285,714}$$
One last interesting thing about this fraction: $$\begin{align}\frac{1}{7}&=0.142857142857...\\&=0.14+0.0028+0.000056+0.00000112+0.0000000224+0.000000000448+\cdots\\&=\frac{14}{100}+\frac{28}{100^2}+\frac{56}{100^3}+\frac{112}{100^4}+\frac{224}{100^5}+\cdots+\frac{7\cdot 2^n}{100^n}+\cdots\\&=\frac{7}{50}+\frac{7}{50^2}+\frac{7}{50^3}+\frac{7}{50^4}+\frac{7}{50^5}+\cdots+\frac{7}{50^n}+\cdots\\&=\sum_{k=1}^\infty \frac{7}{50^k}.\end{align}$$Each term is double the prior term shifted two places to the right.

Reference
Definition of cyclic sum
Examples

More to explore
The Alluring Lore of Cyclic Numbers by Michael W. Ecker