Sequences and series

PLZ EXPLAIN ME WITH AN EXMAPLE . sequenoes equences Definition : A sequence (a,) is called infinitely large if vK e R3 n, e N a la K Ynzn ntinitely large sequences repi N ch that Edunde As an example, we show that the sequence ((-1) n) is infinitely large. Indeed, for any number K, we can find n, such that I(-1)" nl > K vn2 n Examplesa Th To this end, we solve the inequality n > K, and n> K. t Let n= RK] + 1, where [c] is the integer part of c. Then fornèn, we obtain. n2n >k n >K -1)" n'| > K. 3 Howeve From above Definition it follows that any infinitely large sequence is unbounded. Heue converse is not true: there exist unbounded sequences that are not infinitely large. For example, such is the sequence {(1 - -1))n}. Definition. A sequence fa) is called infinitely small if lim a, = 0, n- that is for any &> 0 there exists n. such that a 0 let us fin Ine In& +1, n In(lgl)< In e n> Inla n.ina where Ine < 0, and In lq| < 0, since e< 1, and lq| < 1. Thus, for nen, we have In & nzn In a