By Patrick M. Fitzpatrick

Simply grasp the elemental options of mathematical research with complicated CALCULUS. awarded in a transparent and easy method, this complicated caluclus textual content leads you to an exact figuring out of the topic through supplying you with the instruments you must be successful. a wide selection of routines is helping you achieve a real knowing of the fabric and examples display the importance of what you study. Emphasizing the team spirit of the topic, the textual content indicates that mathematical research isn't a suite of remoted proof and methods, yet really a coherent physique of information.

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Then we need to find an index N such that for all indices n ~ N. 3) Observe that for every index n. 3) holds for this choice of N. ~ N. • 28 ADVANCED CALCULUS A Comparison Criterion for Convergence We now establish a very useful criterion for showing that a sequence converges. This criterion is so frequently used in the remainder of the book that we name it the Comparison Lemma. 9 The Comparison Lemma Let the sequence {an} converge to the number a. 4) Proof Let E > 0. We need to find an index N such that Ibn- bj < E for all indices n ~ N.

We leave it as an exercise for the reader to determine a 20 and a 30 and also show that there are infinitely many indices n for which an = 1/2. • 1 A sequence cannot be formally determined from explicit knowledge of the initial terms of the sequence, but, as in this example, the construction of the sequence can sometimes be clearly indicated by such a display. 4 Let r be any number. Define the sequence {sn} by for every index n. If r = 1, then sn = n, while if r =f. 5 Define the sequence {sn} by for every index n.

Indeed, + an-2b + ... + abn-2 + bn-1) =an+ an-1b + an-2b2 + ... + a2bn-2 + abn-1 (a- b)(an-1 - an-1b- an-2b2- ... - a2bn-2- abn-1 - bn In the Difference of Powers Formula, if we take a = 1, set b = r =j:. 1, and replace n by n + 1, then after division by 1 - T we obtain the following important identity. The Geometric Sum Formula For any natural number n and any number r =j:. 1, 1+r 1 - rn+1 -+ r 2 + · · · + rn = -1-r This formula is the essential tool underlying the frequent possibility of expressing functions as power series that we will consider in Chapters 8 and 9.