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

The value of
\[ \lim_{n \to \infty} \sum_{k=2}^n \frac{\sqrt{n} + 1 - \sqrt{n}}{k(\ln k)^2} \]
is equal to
Let (xn) be a sequence of real numbers. Consider the set P = {n\(\isin\N:x_n\gt x_m\) for all \(m\isin\N\) with \(m\gt n\)}. Then which of the following is/are true?
The sum of the series
\(∑^∞_{ n=1} \frac{1}{(4n-3)(4n+1) }\) 
is equal to_____.(Round off to two decimal places)
Let r be the radius of convergence of the power series
\(\frac{1}{3}+\frac{x}{5}+\frac{x^2}{3^2}+\frac{x^3}{5^2}+\frac{x^4}{3^3}+\frac{x^5}{5^3}+\frac{x^6}{3^4}+\frac{x^7}{5^4}+...\)
Then the value of r2 is equal to _________. (Rounded off to two decimal places)

Let \( a_n = \dfrac{b_{n+1}}{b_n} \), where \( b_1 = 1 \), \( b_2 = 1 \), and \( b_{n+2} = b_n + b_{n+1} \), \( n \in \mathbb{N} \). Then \( \lim_{n \to \infty} a_n \) is