By S. Zaidman.
Ch. 1. Numbers --
ch. 2. Sequences of genuine numbers --
ch. three. limitless numerical sequence --
ch. four. non-stop capabilities --
ch. five. Derivatives --
ch. 6. Convex capabilities --
ch. 7. Metric areas --
ch. eight. Integration.
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Extra info for Advanced calculus : an introduction to mathematical analysis
We have an > OVn G N; next, a\ < 2 and if we assume that an < 2 it follows that an+1 = y/2 + an < y/2 + 2 = 2. Thus, a n < 2 Vn G N. Then, we also have: a\ < a 2 < . . an < a n + i In fact, a 2 > A/2 = ai is obvious. If we assume that ftn+i — &n ^ 0 we derive: &n_f-2 — ^n+i — (2 -f a n + i ) - (2 + an) = an+1 - an > 0. Hence a n + 2 - a n + i > 0 too. Therefore lim an = L > 0 will exist. From relation a n + i = y/2 + a n , we get L = \/2 + L, hence L 2 - L - 2 = 0. As L > 0, it follows that L = 2.
N i " kk)' n Partial sums S P"' Note *(FFP7 tne are ex identity: Pressed In the second sum we put k+p = /, and we get J ] k(k+v) = v as £*- Lfc=i p+n I Z^ / Z=p+1 J pL x ^ 2 ^ n p+1 p+2 p+nJ Let us take n > p. Then we can write: E-^ k(k 1+ p) 1 p 1 1 2+ "" p 1 + 1 n + 1 p+1 1 1 p~+T + P + 2 1 1 -p n+ 1 n+2 1 2 Accordingly: £ j ^ ) = ^ 1 n 1_ p+2 1 n+ 1 1 n+ p 1 = 5 n (nth-partial sum). n+ p 5 n = ±(1 + \ + £). )(*+ 2) 2 ^ * fc + 2 ■ ) x E}k k=l u k=l 1 1 1 + l~k +2 ) = n+ 1 ^r-—v 1 = n+2 1 ^—\ 1 n+1 ^—s 1 2 ^ k ~ 2 ^ k ~ ^ k k=2 1 k=l 1 1k=3 n + 2' 2 n+2 n+2 ^—\ 1 + 2^k k=3 Infinite Numerical Series Hence sn->|(l 59 + i ) - | = i a s n - > o o .
UUU/U < q2anQ\-- \ l£ JLW1 ft/ ^ _ I t/(J . and anQ+v < Infinite Numerical Series 53 As 53(tf')r is convergent, we obtain 53 an0+P convergent, hence 53 a n is 1 1 P=I convergent. 4 (II) and for r > r' > 1 we find n\ such that ^ ± 1 > r' for n > n\. As previously we find that, \/p G N ani+p > (r')vani CO CO and because of divergence of the series 53( r ') p we obtain that 53 a n is also 1 1 divergent. Corollary. If an > 0 for all n and if lim ^ ± CO exists, then the series 53 an l converges if the limit is < 1 and diverges if the limit is > 1.