By J. C. Burkill

ISBN-10: 0521294681

ISBN-13: 9780521294683

This simple path in accordance with the assumption of a restrict is meant for college students who've obtained a operating wisdom of the calculus and are prepared for a extra systematic remedy which additionally brings in different proscribing techniques, akin to the summation of limitless sequence and the growth of trigonometric features as energy sequence. specific consciousness is given to readability of exposition and the logical improvement of the subject material. lots of examples is incorporated, with tricks for the answer of lots of them.

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Extra resources for A First Course in Mathematical Analysis

Sample text

Both sn and s,1tend to the same limit s. 53 (i), lim u„ exists and lim u„ = s—s = O. I Note carefully that the converse of (3) is false. The example un, = 1In shows that it is possible to have lim un = 0 and Zun divergent. In other words (see exercise 2 (g), 1) The condition lim un = 0 is necessary but not sufficient for the convergence of Zun. (4) If Eunis convergent, then so is any series whose terms are obtained by bracketing the terms of Zunin any manner, and the two series have the same sum.

Un or, more shortly, sn =Fur. If, as n co, sntends to a finite limit s, we say that the infinite series urF u2 + + or E unconverges (or, is convergent) and that s is its sum. The number unis the nth term of the infinite series and sn is the sum of the first n terms. ) You should note carefully that, when applied to infinite series, the meaning of the word sum has been widened from its use in algebra. Hitherto it has meant the number which is got by adding the numbers contained in some given finite set.

Continuous functions The reader will have acquired from examples the impression that the common functions can reasonably be called continuous, though some of them may present discontinuities for particular values of x. 3 as being continuous except at x = 2 and x = 4. He would think of a function as continuous so long as its graph can be drawn without taking the pencil off the paper. We must now refine these rough ideas into analytical concepts. The reader, after reflection, will (we hope) agree that for a function f to be continuous at a value x = c he requires that (1) f(c) is defined, (2) as x approaches c, the value of f(x) approaches f(c).

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