By L. Bieberbach

Translated from the fourth German variation by means of F. Steinhardt, with an multiplied Bibliography.

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**Example text**

34 THE CALCULUS On the other hand, Theorem II requires a proof. This will be our first general proof to be given without preparation on particular examples. But it is time that we try our hand at it. First some preliminary remarks: The sequence VI, Vi, V3,V4,... , ... , is bounded above but is not monotonic. This is merely to show what we mean by requiring the sequence to be both monotonic and bounded. Since the sequence is to be bounded, there is an M such that all s; ~ M. This M need not be an integer.

However, according to the continuity axiom in its original formulation, m can be chosen so great that m times the given quantity x-I will exceed any preassigned number, such as the reciprocal 1/ E of the preassigned arbitrarily small E mentioned in the lemma above. Then, a fortiori, x'" > l/E, and, consequently, a'" = l/x'" < Ej and it was the existence of such an exponent which we had to demonstrate. 24 THE CALCULUS Now we have assembled all the tools needed to proceed with the summation of infinitely many terms.

We did not add up all terms of the series; we added the first n of them and then let n increase. On this process we shall base our general definition: DEFINITION. The infinite series al + a2 + ... is called conoergeniif the sequenceof its "pa,tial sums" SI ==ai, S2==al + a2, ... , Sn= al + a2 + ... + an is conoergent;and the limit s = lim Sn = lim (a, + ... +a n) D----i(X) is called"the sum" of the series: a, D-+(X) + a2 + ... = s. The series 1 + ! + i + ... , 1,.... Both are convergent. But the series has the sum 2; the sequence has the limit O.