List of top Mathematics Questions on Sequences and Series of real numbers

Let \(\{a_n\}_{n\geq 1}\) be a sequence of real numbers given by

Let \(f:\mathbb{N}\to\{1,2,3,\ldots,100\}\) be an onto function. Consider the following statements.

\[ \lim_{n\to\infty}\left(\frac{n^2+1}{\sqrt{n^6+1}}+\cdots+\frac{n^2+n}{\sqrt{n^6+n}}\right) = \underline{} \] rounded off to one decimal place.

Which of the following statements is/are TRUE?

For each positive integer \( n \), let \[ x_n = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} - \log n \]
and \[ y_n = \int_1^n \frac{\cos t}{t^2} \, dt. \]
Which ONE of the following statements about the sequences \( (x_n) \) and \( (y_n) \) is TRUE?