List of top Mathematics Questions on Sequences and Series asked in CPET

Consider the sequences \(\{a_n\}\), \(\{b_n\}\), \(\{c_n\}\) defined for \(n \in \mathbb{N}\) by \[\text{(I) } a_n = n^2 \sin\left(\frac{1}{n}\right), \qquad \text{(II) } b_n = 1 + \frac{(-1)^n}{n}, \qquad \text{(III) } c_n = n\cos\left(\frac{1}{n}\right).\] Pick out the correct option.

Let the limits of the sequences \(\{x_n\}_{n\ge1}\) and \(\{y_n\}_{n\ge1}\) be \(\lambda\) and \(\lambda^3\) respectively. If the interleaved sequence \(x_1, y_1, x_2, y_2, x_3, y_3, \dots\) has a limit, then its value is ____.

Which of the following assertions is correct? \[\text{(I) } \lim_{n\to\infty}\left(1+\frac1n\right)^{n^2}e^{-n} = 1 \qquad \text{(II) For } 0 < a < 1 < b,\ \lim_{n\to\infty}\left(a+\frac{b}{n}\right)^n = e^{b/a}\]

What is the value of \(a_{32}\) from the recurrence relation: \(a_1 = 4,\ a_n = 5n + a_{n-1}\ (n \ge 2)\)?