Mathem uences of real numbers infinitely large sequences represent an important subset of unbounded sequences. definition : a sequence {a,} is called infinitely

Mathem uences of Real Numbers Infinitely large sequences represent an important subset of unbounded sequences. Definition : A sequence {a,} is called infinitely large if VK e R 3 n, e N such that la> K Vn2 n As an example, we show that the sequence {(-1)" n3} is infinitely large. Indeed, for any number K, we can find n, such that |(-1)" n°| > K Yn 2 n To this end, we solve the inequality n3> K, and n> K.t Let n = [VK] + 1, where [c] is the integer part of c. Then forn2 n, we obtain. n2n, >K n°>K -1" n°|>K. From above Definition it follows that any infinitely large sequence is unbounded. However, the converse is not true: there exist unbounded sequences that are not infinitely large. For example, such is the sequence {(1 - (-1")n}.