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Calculus Gallery William Dunham Paperback. More than three centuries after its creation, calculus remains a dazzling intellectual achievement and the gateway into higher mathematics. This book charts its growth and development by sampling from the work of some of its foremost practitioners, beginning with Isaac Newton and Gottfried Wilhelm L. Used: Acceptable.

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No International ShippingGood condition. Quick ship! Used book doesn't include any access code, CD or other supplements. For returns contact us. Very Good. Calculus Gallery Dunham Paperback. Theorem: The harmonic series diverges. Proof: Choosing an arbitrary whole number N, Jakob sought to remove from the beginning of the harmonic series finitely many consecutive terms whose sum is equal to or greater than 1. From what remained, he extracted a finite string of consecutive terms whose sum equals or exceeds another unity.

He continued in this fashion until N such strings had been removed, making the sum of the entire harmonic series as least as big as N. Because N was arbitrary, the harmonic series is infinite.

This procedure, taken almost verbatim from Jakob's original, is fine provided we can always remove a finite string of terms whose sum is I or more. To complete the argument, Bernoulli had to demonstrate that this is indeed the case. He thus assumed the opposite, stating, "If, after having removed a number of terms, you deny that it is possible for the rest to surpass unity, then let 11a be the first remaining term after the last removal.

In this way Jakob established that, starting at any point of the harmonic series, a finite portion of what remained must sum to one or more. The resulting sum can therefore be made greater than any preassigned number, and so the harmonic series diverges. This was a clever argument. Its significance was not lost on Jakob, who emphasized that, "The sum of an infinite series whose final term vanishes is perhaps finite, perhaps infinite" [14].

The harmonic series stands as the great example to illustrate this point. Of equal interest were well-behaved infinite series having finite sums. Starting with the geometric senes and cleverly modifYIng the outcome, Jakob proceeded until he could calculate the exact values of some nontrivial series. We consider a few of these below. First he needed the sum of an infinite geometric progression.

So too were the "figurate numbers," families of integers related to such geometrical entities as triangles, pyramids, and cubes. As an example we have the triangular numbers 1, 3, 6, 10, 15, It can. Of course, the square numbers 1,4,9, 16,25, For instance, he wished to evaluate such series as I or k I ;. Jakob attacked the problem by building from the simple to the complicated-always a good mathematical strategy. Following his arguments, we begin with an infinite series having the natural numbers as numerators and a geometric progression as denominators [15].

Proof: This follows easily because 3 Hence 1 - 4.!. After such successes, Jakob Bernoulli may have begun to feel invincible. He could show, using what we now recognize as the comparison test, that the series converges to some number less than 2, but he was unable to identify it. Swallowing his pride, Jakob included this plea in his Tractatus: "If anyone finds and communicates to us that which has thus far eluded our efforts great WIll be our gratitude" [16].

Jakob Bernoulli was a master of infinite senes. His brother Johann, equally gifted, had his own research interests. Among these was what he called the "exponential calculus," which will be our next stop. At that time in mathematical publishing, higher powers were typeset as they are today, but the quadratic x 2 was often written xx.

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Also, in the interest of full disclosure, we observe that Bernoulli denoted the natural logarithm ofx by Ix. Fortunately,Johann corrected his error. With these preliminaries behind him, Johann promised to apply principles "first invented by me" to reap a rich harvest of knowledge "incrementing this new infinitestimal calculus WIth results not previously found or not WIdely known" [19].

For an arbitrary point F on the curve, Johann sought the subtangent, that is, the length of segment LE on the x-axis beneath the tangent line. This occurs when the tangent line is horizontal or, equivalently, when the subtangent is infinite. His reasoning was fine, but the form of his answer seems, to modern tastes, less than optimal. Johann was just warming up. That is, he wanted the value of Jo x dx. Remarkably enough, he found what he was seeking [21].

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The argument required two preliminaries. He proceeded recursively to generate the table shown on this page.

In this fashion, we replicate Bernoulli's list of integrals. Along Wlth the exponential series in 3 , this was the key to solving his curious problem. Today, we would apply l'Hospital's rule a most fitting choice!

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He noted, "This wonderful series converges so rapidly that the tenth term contnbutes only a thousandth of a millionth part of unity to the sum" [23]. As the examples in this chapter should make clear, Jakob and Johann Bernoulli were worthy disciples of Gottfried Wilhelm Leibniz.

In their hands, his calculus became, as we might say today, "user-friendly" The brothers left the subject in a more sophisticated yet much more understandable state than they found it.

The student's name was Leonhard Euler, and we sample his work next. Euler Leonhard Euler In any accounting of historys greatest mathematicians, Leonhard Euler stands tall. With broad and inexhaustible interests, he revolutionized mathematics, extending the boundaries of such weUestablished subdisciplines as number theory, algebra, and geometry even while giving birth to new ones like graph theory, the calculus of variations, and the theory of partitions. When in scholars began publishing his collected works, the Opera omnia, they faced a daunting challenge.

Today, after more than seventy volumes and 25, pages in print, the task is not yet complete. This enonnous publishing proJect, consuming the better part of a century, bears WltnesS to a mathematical force of nature. That force was especially evident in analysis.

## The Calculus Gallery: Masterpieces from Newton to Lebesgue

Among Eulers collected works are eighteen thick volumes and nearly pages on the subject. These include landmark textbooks on functions , differential calculus , and integral calculus , as well as dozens of papers on topics 52 53 EULER ranging from differential equations to infinite series to elliptic integrals. As a consequence, Euler has been descnbed as "analysis incarnate" [1].