If 1+ √3-√2 2 √3+ 5-2 √6 18+ 9 √3-11 √2 36 √3+ 49-20 √6 180+… upto ∞=2+ (√ b a+1 ) log _e ( a b ), where a and b are integers with gcd( a, b)=1, then…

$$\begin{aligned} & S=1+\frac{x}{2 \sqrt{3}}+\frac{x^2}{18}+\frac{x^3}{36 \sqrt{3}}+\frac{x^4}{180}+\ldots \infty \\ & \text { Put } \frac{x}{\sqrt{3}}=t, \text { where } x=\sqrt{3}-\sqrt{2} \\ & S=1+\frac{t}{2}+\frac{t^2}{6}+\frac{t^3}{12}+\frac{t^4}{20}+\ldots \\ & S=1+t\left(1-\frac{1}{2}\right)+t^2\left(\frac{1}{2}-\frac{1}{3}\right)+t^3\left(\frac{1}{3}-\frac{1}{4}\right)+t^4\left(\frac{1}{4}-\frac{1}{5}\right) \\ & S=\left(1+t+\frac{t^2}{2}+\frac{t^3}{3}+\frac{t^3}{4}+\ldots\right)-\left(\frac{t}{2}+\frac{t^2}{3}+\frac{t^3}{4}+\frac{t^4}{5}+\ldots\right) \\ & S=\left(t+\frac{t^2}{2}+\ldots\right)-\frac{1}{t}\left(t+\frac{t^2}{2}+\frac{t^3}{3}+\ldots\right)+2 \\ & S=2+\left(1-\frac{1}{t}\right)(-\log (1-t))=\left(\frac{1}{t}-1\right) \log (1-t)+2 \\ & S=2+\left(\frac{\sqrt{3}}{\sqrt{3}-\sqrt{2}}-1\right) \log \left(1-\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}}\right) \\ & S=2+\left(\frac{\sqrt{2}}{\sqrt{3}-\sqrt{2}}\right) \log e \frac{\sqrt{2}}{\sqrt{3}} \\ & S=2+\frac{(\sqrt{6}+2)}{2} \log e \frac{2}{3}=2+\left(\sqrt{\frac{3}{2}}+1\right) \log e \frac{2}{3} \\ & a=2, b=3 \\ & 11 a+18 b=11 \times 2+18 \times 3=76\end{aligned}$$​S=1+23​x​+18x2​+363​x3​+180x4​+…∞ Put 3​x​=t, where x=3​−2​S=1+2t​+6t2​+12t3​+20t4​+…S=1+t(1−21​)+t2(21​−31​)+t3(31​−41​)+t4(41​−51​)S=(1+t+2t2​+3t3​+4t3​+…)−(2t​+3t2​+4t3​+5t4​+…)S=(t+2t2​+…)−t1​(t+2t2​+3t3​+…)+2S=2+(1−t1​)(−log(1−t))=(t1​−1)log(1−t)+2S=2+(3​−2​3​​−1)log(1−3​3​−2​​)S=2+(3​−2​2​​)loge3​2​​S=2+2(6​+2)​loge32​=2+(23​​+1)loge32​a=2,b=311a+18b=11×2+18×3=76​