By S. Zaidman.
Ch. 1. Numbers --
ch. 2. Sequences of actual numbers --
ch. three. endless numerical sequence --
ch. four. non-stop services --
ch. five. Derivatives --
ch. 6. Convex capabilities --
ch. 7. Metric areas --
ch. eight. Integration.
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Sample text
Fc=i fc=i We shall use the ratio: (D'Alembert) test for convergence. The quotient 2=±i equals: $±$r • £ = („ + 1 ) ( ^ ) « • ^ = ( T ^ F ' W e k»™ that: lim T^pW exists and = \. - m T = sunrr > > 2 Thus * < * and the series above is convergent" ' (6) We shall prove that if ak > 0, VA: G N, the series f > f c and £ > * o o (1 + a , ) - 1 are convergent or divergent simultaneously. Let us assume convergence oi 2^afc- -^rom relation: a/c(l + a^j < a^ oo we derive convergence of V a n ( l + a n ) _ 1 .
N + - + 1 ^! We also have the obvious lower estimate: n! = 1 • 2 . . n > 2 n _ 1 , n G N, hence x n < l + H - | + . . yhr + • • • + 2 ^ r = 1 + T ^ r < 1 + x = 3, Vn G N. (Here we applied the elementary formula: 1 + x + x2 + .. xn = 1~1X_X which can be obtained as follows: put s = 1 4- x 4- . . xn. Thus xs = x 4- x2 4- . . xn+1, hence (1 — x)s = 1 — xn+1). Let us now note a slight extension of Theorem 1. Theorem 3. A monotone increasing sequence is either convergent (to a real number) or is convergent to +oo.
Let (a n ), (bn) be two sequences, where an -^ a and 6 n —> +oo. It follows that (an + 6n) —► +oo. In fact, take any M > 0, we have 6n > M for n > n, take also e = 1, we get a n > a — 1 for n > l\ thus, for n > max (n,I) we obtain an + bn > M -ha — 1. Finally, if Mi > 0 is given, take M such that M + a - 1 > Mx. Example 10. Let (a n )i° be a sequence, such that \an — a n + i | < Vn G N with some c e (0,1). Show that it is a convergent sequence. Acn, We shall see that it is a Cauchy sequence, let us evaluate \an — a m |, assume m > n and put p = m — n; we have | ^ n — Gn+pl < |^n ~ &n+l \ + | a n + l ~ «-n+2| + .