Elementary Methods in the Analytic Theory of Numbers by A. O Gelfond, Yu. V. Linnik, L. J. Mordell

By A. O Gelfond, Yu. V. Linnik, L. J. Mordell

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1 x 2 x 3 is greater than 3). * 'Kathryn, 4, learnt the meaning of n\, and loves to recite it when asked. Infinity and Infinite Series 7 The Harmonic Series 1 1 1 1 1 1 1 1 2 3 4 5 6 7 —^ oo See Proof 14 -> (page 165) IF IT IS obvious that the sum of integers tends to infinity, what about the sum of their reciprocals (opposite page)? First of all, let us note that the bigger the integer, the smaller is the reciprocal. So the individual reciprocals would get smaller and smaller, and tend to zero, as the integers themselves get larger and larger and tend to infinity.

What an "eureka" moment. What serendipity! And the proof is simple. Obviously since the Liebniz-Gregory equation was discovered some three hundred years ago, thousands (probably millions) of mathematicians must have discovered this equation too. It's just that I have not seen it before in the mathematics literature that I've read so far. This incident shows that there are still "eureka" moments when one discovers something by one's own effort. It does not matter, as in the case of Liebniz, that someone else had discovered it before.

Then they start escalating. "I have 4", "I have 10". "I have 20", "I have 100", and so on. In next to no time, they begin to use bigger numbers, such as thousands, millions, and billions. Soon they run out of all the names of numbers that they know, and they move on to other subjects. The concept of infinity is unknown to children and even to most adults. Indeed it was unknown to many earlier mathematicians who were confused by it, and avoided thinking or talking about the concept. This lasted until about the end of the 16th century.

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