Problem 7

(Problem posed by Department of Mathematics, Sarada Vilas College)

Problem: Show that the sequence \(\large{\{cos(\pi x n!)\}_{n=1}^{\infty}}\) converges to \(1\) for all rarional numbers \(x\). Surprisingly, there are infinitely many irrational numbers \(x\) for which the sequence converges (in fact to \(1\)). Can you find at least one such irrational number? (If possible, infinitely many).

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Note: A sequence is an ordered arrangement of infinite numbers. For example: \(1, 4, 7, 11, 15, \dots, 3n-2, \dots\). \(3n-2\) simply gives the rule for computing the number which should come at \(n^{th}\) place. Vaguely, we say that a sequence converges to a number \(L\), called the limit, if the values of its terms become closer to $L$ as the sequence grows. A sequence which does not converge to any number is said to be divergent sequence. For example, \( 1, \frac{1}{2}, \frac{1}{3}, \dots, \frac{1}{n}\dots \) converges to \(0\), while \(1,2,3,\dots, n\dots \) does not converge to any number.

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