SELECTED CHAPTERS FROM ALGEBRA I. R. Shafarevich Preface

8 I. R. Shafarevichc n =1,thena n = n and the sequence a is unbounded. We considered a less trivialexample in Section 2 of Chapter IV: in the sequence c all c n =1=n. We sawthatinthat case the sequence a is also unbounded. But if we cancheck that the sequencea of sums is bounded, then according to Theorem 1 it has a unique limit . Thislimit is called the sum of the sequence (c 1 c 2 ...c n ...), which is denoted byc 1 + c 2 + + c n + = :Sometimes the innite sum c is called a series and its sum|the sum of the series.If the sequence of sums a n is bounded, then, as we have seen, the sum of theseries c 1 + c 2 + + c n + exists. If it is unbounded, then we say that the sumof the series does not exist. Hence, Lemma 1, Section 2, Chapter IV, states thatthe sum of the series 1 + 1 2 + 1 + does not exist.3Consider an example. Let a nonnegative number a, lessthan1,begiven, andlet c =(1aa 2 ...a n ...). Then a n =1+a + a 2 + + a n;1 (in the n-th placein the sequence c there appears a n;1 ). The sum 1+a + a 2 + + a n;1 can beevaluated using the formula for the sum of a geometric progression|formula (12)of Chapter I:(1) a n =1+a + a 2 + + a n;1 = 1 ; an1 ; a = 11 ; a ; an1 ; a :We have seenthata n ! 0whenn !1, wherefrom it follows immediately thata n1 ; a ! 0 when n !1. Thus, formula (1) gives that a n ! 1 . We can write1 ; athis as:(2) 1+a + a 2 + + a n + = 1 for a