By J. C. Burkill

This easy path according to the assumption of a restrict is meant for college kids who've received a operating wisdom of the calculus and are prepared for a extra systematic remedy which additionally brings in different proscribing approaches, comparable to the summation of limitless sequence and the growth of trigonometric features as strength sequence. specific recognition is given to readability of exposition and the logical improvement of the subject material. a good number of examples is integrated, with tricks for the answer of lots of them.

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**Example text**

We have further If n is a positive integer, the function xn is continuous for all values of x; x-n is continuous except for x = 0. This can be proved directly (by proving that xi' — cn is small if x— c is small) or alternatively by induction, applying the theorem about the product of continuous functions to xn-1and x. We can now build up sums of multiples of powers of x to give Any polynomial is continuous for all values of x. 5 A quotient of two polynomials is continuous for all values of x for which the denominator is not zero.

Oscillates, but the series (1 1)+0 0+ ... converges to sum 0. (5) If un0 for every n, then Ell," either converges or diverges to +0o. A necessary and sufficient condition for convergence is that co N there exists K such that Din< K for all N; and then Zun < K. 6. (6) If, for every n, - (i) un - 0, v > 0, (ii) un< Kv„, where K is constant, (iii) Evnconverges, then Zunconverges. Also Eun < KEvn. The reader should deduce this from (5) and should formulate a corresponding result for divergence. We shall take up in chapter 5 the more systematic investigation of infinite series.

54. 53 applied to the product of the two sequences snand 1/4. We wish therefore to prove that 1 1 t— that is to say t,, t ' t„ t ' is a null sequence. We can choose N such that, for all n > N, Itn i > I tl 1 2 and so itntl < 1t12. 52 shows that (t — t„)/t,,t is a null sequence. I Exercises 2 (c) 172. 1. Discuss the behaviour as n oo of the sequences whose nth terms are Notes on these exercises are given on p. n— 3 \3 13n+1\2 13n +11 ' ln-3I ' n(n-1) (n-2) (n — 3) (n — 4) 2. • ++ ba where p and q are positive integers.