Mathematics - Sequence & Series Question with Solution | TestHub

$\begin{array}{rl}& S=1+\frac{x}{2\sqrt{3}}+\frac{x^2}{18}+\frac{x^3}{36\sqrt{3}}+\frac{x^4}{180}+\dots \infty \\ & \text{ Put }\frac{x}{\sqrt{3}}=t,\text{ where }x=\sqrt{3}-\sqrt{2}\\ & S=1+\frac{t}{2}+\frac{t^2}{6}+\frac{t^3}{12}+\frac{t^4}{20}+\dots \\ & S=1+t\left(1-\frac{1}{2}\right)+t^2\left(\frac{1}{2}-\frac{1}{3}\right)+t^3\left(\frac{1}{3}-\frac{1}{4}\right)+t^4\left(\frac{1}{4}-\frac{1}{5}\right)\\ & S=\left(1+t+\frac{t^2}{2}+\frac{t^3}{3}+\frac{t^3}{4}+\dots \right)-\left(\frac{t}{2}+\frac{t^2}{3}+\frac{t^3}{4}+\frac{t^4}{5}+\dots \right)\\ & S=\left(t+\frac{t^2}{2}+\dots \right)-\frac{1}{t}\left(t+\frac{t^2}{2}+\frac{t^3}{3}+\dots \right)+2\\ & S=2+\left(1-\frac{1}{t}\right)(-\log (1-t))=\left(\frac{1}{t}-1\right)\log (1-t)+2\\ & S=2+\left(\frac{\sqrt{3}}{\sqrt{3}-\sqrt{2}}-1\right)\log \left(1-\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}}\right)\\ & S=2+\left(\frac{\sqrt{2}}{\sqrt{3}-\sqrt{2}}\right)\log e\frac{\sqrt{2}}{\sqrt{3}}\\ & S=2+\frac{(\sqrt{6}+2)}{2}\log e\frac{2}{3}=2+\left(\sqrt{\frac{3}{2}}+1\right)\log e\frac{2}{3}\\ & a=2,b=3\\ & 11a+18b=11\times 2+18\times 3=76\end{array}$