List of top Statistics Questions on Sequences and Series of real numbers asked in IIT JAM MS

Let \(\{a_n\}_{n \ge 1}\) be a sequence of real numbers such that \(a_n \ge 1\), for all \(n \ge 1\). Then, which of the following conditions imply the divergence of \(\{a_n\}_{n \ge 1}\)?

Let \[ S = \sum_{k=1}^{\infty} (-1)^{k-1}\frac{1}{k}\left(\frac{1}{4}\right)^k, \quad T = \sum_{k=1}^{\infty} \frac{1}{k}\left(\frac{1}{5}\right)^k. \] Then, which of the following statements is TRUE?

If the series \(\sum_{n=1}^{\infty} a_n\) converges absolutely, then which of the following series diverges?

Let the Taylor polynomial of degree 20 for \( \frac{1}{(1-x)^3} \) at \( x = 0 \) be

\[ \sum_{n=0}^{20} a_n x^n \]

Then \( a_{15} \) equals

Let \(\{x_n\}_{n \geq 1}\) be a sequence of positive real numbers such that the series \(\sum_{n=1}^{\infty} x_n\) converges. Which of the following statements is (are) always TRUE?

The sum \[ \sum_{n=4}^{\infty} \frac{6}{n^2 - 4n + 3} \quad \text{equals} \]

Let \( u_n = \frac{(4 - n)}{n} \), \( n \in \mathbb{N} \), and let \( l = \lim_{n \to \infty} u_n \).
Which of the following statements is TRUE?

Let \( X_1, X_2, X_3 \) be i.i.d. discrete random variables with the probability mass function
\[ p(k) = \left( \frac{2}{3} \right)^{k-1} \left( \frac{1}{3} \right), \quad k = 1, 2, 3, \dots \] Let \( Y = X_1 + X_2 + X_3 \). Then \( P(Y \geq 5) \) equals