Correct answer: 76
\(\mathrm{S=1+\frac{x}{2 \sqrt{3}}+\frac{x^{2}}{18}+\frac{x^{3}}{36 \sqrt{3}}+\frac{x^{4}}{180}+\ldots \infty}\)
Put \(\frac{x}{\sqrt{3}}=t\), where \(x=\sqrt{3}-\sqrt{2}\)
\(\mathrm{S}=1+\frac{\mathrm{t}}{2}+\frac{\mathrm{t}^{2}}{6}+\frac{\mathrm{t}^{3}}{12}+\frac{\mathrm{t}^{4}}{20}+\ldots\)
\(\mathrm{S}=1+\mathrm{t}\left(1-\frac{1}{2}\right)+\mathrm{t}^{2}\left(\frac{1}{2}-\frac{1}{3}\right)+\mathrm{t}^{3}\left(\frac{1}{3}-\frac{1}{4}\right)+\mathrm{t}^{4}\left(\frac{1}{4}-\frac{1}{5}\right)\)
\(\mathrm{S}=\left(1+\mathrm{t}+\frac{\mathrm{t}^{2}}{2}+\frac{\mathrm{t}^{3}}{3}+\frac{\mathrm{t}^{3}}{4}+\ldots\right)-\left(\frac{\mathrm{t}}{2}+\frac{\mathrm{t}^{2}}{3}+\frac{\mathrm{t}^{3}}{4}+\frac{\mathrm{t}^{4}}{5}+\ldots\right)\)
\(\mathrm{S}=\left(\mathrm{t}+\frac{\mathrm{t}^{2}}{2}+\ldots\right)-\frac{1}{\mathrm{t}}\left(\mathrm{t}+\frac{\mathrm{t}^{2}}{2}+\frac{\mathrm{t}^{3}}{3}+\ldots\right)+2\)
\(\mathrm{S}=2+\left(1-\frac{1}{\mathrm{t}}\right)(-\log (1-\mathrm{t}))=\left(\frac{1}{\mathrm{t}}-1\right) \log (1-\mathrm{t})+2\)
\(\mathrm{S}=2+\left(\frac{\sqrt{3}}{\sqrt{3}-\sqrt{2}}-1\right) \log \left(1-\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}}\right)\)
\(\mathrm{S}=2+\left(\frac{\sqrt{2}}{\sqrt{3}-\sqrt{2}}\right) \log \mathrm{e} \frac{\sqrt{2}}{\sqrt{3}}\)
\(\mathrm{S}=2+\frac{(\sqrt{6}+2)}{2} \log \mathrm{e} \frac{2}{3}=2+\left(\sqrt{\frac{3}{2}}+1\right) \log \mathrm{e} \frac{2}{3}\)
\(\mathrm{a}=2, \mathrm{~b}=3\)
\(11 \mathrm{a}+18 \mathrm{b}=11 \times 2+18 \times 3=76\)
