Expand the expression $\left(x+\frac{1}{x}\right)^{6}$.

Using the Binomial Theorem,the expansion of $\left(x+\frac{1}{x}\right)^{6}$ is given by:
$\left(x+\frac{1}{x}\right)^{6} = \sum_{k=0}^{6} \binom{6}{k} x^{6-k} \left(\frac{1}{x}\right)^{k}$
$= \binom{6}{0}x^{6} + \binom{6}{1}x^{5}\left(\frac{1}{x}\right) + \binom{6}{2}x^{4}\left(\frac{1}{x^{2}}\right) + \binom{6}{3}x^{3}\left(\frac{1}{x^{3}}\right) + \binom{6}{4}x^{2}\left(\frac{1}{x^{4}}\right) + \binom{6}{5}x\left(\frac{1}{x^{5}}\right) + \binom{6}{6}\left(\frac{1}{x^{6}}\right)$
$= 1 \cdot x^{6} + 6 \cdot x^{4} + 15 \cdot x^{2} + 20 \cdot 1 + 15 \cdot \frac{1}{x^{2}} + 6 \cdot \frac{1}{x^{4}} + 1 \cdot \frac{1}{x^{6}}$
$= x^{6} + 6x^{4} + 15x^{2} + 20 + \frac{15}{x^{2}} + \frac{6}{x^{4}} + \frac{1}{x^{6}}$