By Komarchev I.A.
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Additional info for 2-Abolutely summable oeprators in certain Banach spaces
As a preliminary trick, we divide numerator and denominator by the leading power, and rewrite the expression as 3 + (2/n) - l/n 2 5 - (4/n) + (8/n 2 ) • Since l/n -+ 0 and l/n 2 -+ 0, the sum and product statements show that the numerator converges to 3 and the denominator converges to 5. By the quotient statement, the quotient then converges to ~. The most useful criteria for convergence of sequences of real and complex numbers are gathered in the next several theorems. The first criterion is sometimes called the in-between theorem.
Then (n: 1) / (~) --+-1. ~ -1, a# -1, then I(n: 1) I > I(~) I for all n~ O. If Re a> -1 and ais not an integer, then I(n ~ 1) I < I (~) I (d) If Rea (e) for n large. x; n 7. Define Xo = 0, and define by induction Xn+l = + ~ for ~ O. Show that Xn --+ ~. Hint. Show that the sequence is bounded and monotone, and that any limit satisfies x = x2 + ~. 8. Show that if Sn --+ S, then ISn - sn-ll --+ O. 9. Plot each sequence and determine its lim inf and lim sup. 1 (a) Sn = 1 + - + (-It (c) Sn = sin(7fnI4) n (b) Sn = (-nt (d) Sn = xn (x E lR fixed) 10.
33 II 34 Analytic Functions The language of formal mathematics serves to quantify this statement and make it precise. The "small disk" is traditionally given radius £ > O. That "sn lies in the disk" means that ISn - sl < £. That an event "eventually" occurs is translated to the statement that there is N 2:: 1 such that the event occurs for n 2:: N. Thus the translation of the definition of convergent sequence to the language of formal mathematics is as follows. Is" -sl < e for n~4 Formal Definition.
2-Abolutely summable oeprators in certain Banach spaces by Komarchev I.A.