If $1+\frac{\sqrt{3}-\sqrt{2}}{2 \sqrt{3}}+\frac{5-2 \sqrt{6}}{18}+\frac{9 \sqrt{3}-11 \sqrt{2}}{36 \sqrt{3}}+\frac{49-20 \sqrt{6}}{180}+\ldots$ upto $\infty=2+\left(\sqrt{\frac{b}{a}}+1\right) \log _e\left(\frac{a}{b}\right)$, where $\mathrm{a}$ and $\mathrm{b}$ are integers with…

Question

If $1+\frac{\sqrt{3}-\sqrt{2}}{2 \sqrt{3}}+\frac{5-2 \sqrt{6}}{18}+\frac{9 \sqrt{3}-11 \sqrt{2}}{36 \sqrt{3}}+\frac{49-20 \sqrt{6}}{180}+\ldots$ upto $\infty=2+\left(\sqrt{\frac{b}{a}}+1\right) \log _e\left(\frac{a}{b}\right)$, where $\mathrm{a}$ and $\mathrm{b}$ are integers with $\operatorname{gcd}(\mathrm{a}, \mathrm{b})=1$, then $11 \mathrm{a}+18 \mathrm{~b}$ is equal to ______

Aicharya · Accepted Answer

$\begin{aligned} & S=1+\frac{x}{2 \sqrt{3}}+\frac{x^2}{18}+\frac{x^3}{36 \sqrt{3}}+\frac{x^4}{180}+\ldots \infty \ & 	ext { Put } \frac{x}{\sqrt{3}}=t, 	ext { where } x=\sqrt{3}-\sqrt{2} \ & S=1+\frac{t}{2}+\frac{t^2}{6}+\frac{t^3}{12}+\frac{t^4}{20}+\ldots \ & S=1+t\left(1-\frac{1}{2}ight)+t^2\left(\frac{1}{2}-\frac{1}{3}ight)+t^3\left(\frac{1}{3}-\frac{1}{4}ight)+t^4\left(\frac{1}{4}-\frac{1}{5}ight) \ & S=\left(1+t+\frac{t^2}{2}+\frac{t^3}{3}+\frac{t^3}{4}+\ldotsight)-\left(\frac{t}{2}+\frac{t^2}{3}+\frac{t^3}{4}+\frac{t^4}{5}+\ldotsight) \ & S=\left(t+\frac{t^2}{2}+\ldotsight)-\frac{1}{t}\left(t+\frac{t^2}{2}+\frac{t^3}{3}+\ldotsight)+2 \ & S=2+\left(1-\frac{1}{t}ight)(-\log (1-t))=\left(\frac{1}{t}-1ight) \log (1-t)+2 \ & S=2+\left(\frac{\sqrt{3}}{\sqrt{3}-\sqrt{2}}-1ight) \log \left(1-\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}}ight) \ & S=2+\left(\frac{\sqrt{2}}{\sqrt{3}-\sqrt{2}}ight) \log e \frac{\sqrt{2}}{\sqrt{3}} \ & S=2+\frac{(\sqrt{6}+2)}{2} \log e \frac{2}{3}=2+\left(\sqrt{\frac{3}{2}}+1ight) \log e \frac{2}{3} \ & a=2, b=3 \ & 11 a+18 b=11 	imes 2+18 	imes 3=76\end{aligned}$

If $1+\frac{\sqrt{3}-\sqrt{2}}{2 \sqrt{3}}+\frac{5-2 \sqrt{6}}{18}+\frac{9 \sqrt{3}-11 \sqrt{2}}{36…

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