List of top Mathematics Questions on Sequences and Series asked in IIT JAM MA

\( \displaystyle \int_0^1 \left(\sum_{k=1}^{\infty}\frac{(\log_e 2)^k x^{k^2-1}}{(k-1)!}\right)dx = \underline{} \) rounded off to one decimal place.

Let \( a_n = \sqrt{n}, n \geq 1 \), and let \( s_n = a_1 + a_2 + \cdots + a_n \). Then \[ \lim_{n \to \infty} \left( \frac{a_n}{s_n} \Big/ \ln \left( 1 - \frac{a_n}{s_n} \right) \right) = \quad \_\_\_\_\_\_\_\_. \]

\[ \frac{1}{2\pi} \left( \frac{\pi^3}{1 \cdot 3} + \frac{\pi^5}{3 \cdot 5} + \frac{\pi^7}{5 \cdot 7} + \cdots + (-1)^n \frac{\pi^{2n+1}}{(2n-1)! (2n+1)} + \dots \right) = \_\_\_\_ \]

Evaluate the limit \[ \lim_{n \to \infty} \frac{1}{\sqrt{n}} \left( \frac{1}{\sqrt{3} + \sqrt{6}} + \frac{1}{\sqrt{6} + \sqrt{9}} + \cdots + \frac{1}{\sqrt{3n} + \sqrt{3n + 3}} \right). \]