Question

If $1 + \frac{\sqrt{3} - \sqrt{2}}{2\sqrt{3}} + \frac{5 - 2\sqrt{6}}{18} + \frac{9\sqrt{3} - 11\sqrt{2}}{36\sqrt{3}} + \frac{49 - 20\sqrt{6}}{180} + \cdots$ up to $\infty = 2 \left( \sqrt{\frac{b}{a}} + 1 \right) \log_e \left( \frac{a}{b} \right)$, where $a$ and $b$ are integers with $\gcd(a, b) = 1$, then (11a + 18b) is equal to _________.

Answer 1

Given the infinite series:

$S = 1 + \frac{x}{2\sqrt{3}} + \frac{x^2}{18} + \frac{x^3}{36\sqrt{3}} + \frac{x^4}{180} + \dots,$ where $x = \sqrt{3} - \sqrt{2}$.

Step 1: Expressing the Series Let:

$t = \frac{x}{\sqrt{3}} \quad \text{where} \quad x = \sqrt{3} - \sqrt{2}.$

Rewriting the series:

$S = 1 + \frac{t}{2} + \frac{t^2}{6} + \frac{t^3}{12} + \frac{t^4}{20} + \dots.$

Step 2: Using the Known Expansion From known series expansions, we have:

$S = 2 + \sum_{n=1}^\infty \frac{t^n}{n(n+1)}.$

Using the expansion:

$S = 2 + \left(- \log(1 - t)\right) + 2.$

Substituting $t = \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3}}$:

$S = 2 + \log\left(\frac{\sqrt{3}}{\sqrt{3} - \sqrt{2}}\right).$

Step 3: Evaluating Constants Given:

$S = 2 \left(\sqrt{\frac{b}{a} + 1}\right) \log_e\left(\frac{a}{b}\right).$

Comparing terms, we identify:

$a = 2, \quad b = 3.$

Step 4: Calculating the Required Expression

$11a + 18b = 11 \times 2 + 18 \times 3 = 22 + 54 = 76.$

Therefore, the correct answer is $76$.