Leonhard Euler and Infinite Series

Leonhard Euler and Infinite Series

The history of mathematics is valuable as an account of the gradual development of the many current branches of mathematics. It is extremely fascinating and instructive to study even the false steps made by the greatest minds and in this way reveal their often unsuccessful attempts to formulate correct concepts and proofs, even though they were on the threshold of success. Their efforts to justify their work, which we can now appraise with the advantage of hindsight, often border on the incredible.

These features of history are most conspicuous in the work of Leonhard Euler, the key figure in l8th-century mathematics, and one who should be ranked with Archimedes, Newton, and Gauss. Euler’s recorded work on infinite series provides a prime example of the struggles, successes and failures which are an essential part of the creative life of almost all great mathematicians.

Euler first undertook work on infinite series around 1730, and by that time, John Wallis, Isaac Newton, Gottfried Leibniz, Brook Taylor, and Colin Maclaurin had demonstrated the series calculation of the constants e and π and the use of infinite series to represent functions in order to integtate those that could not be treated in closed form. Hence it is understandable that Euler should have tackled the subject. Like his predecessors, Euler’s work lacks rigor, is often ad hoc, and contains blunders, but despite this, his calculations reveal an uncanny ability to judge when his methods might lead to correct results.

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