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e: The Story of a Number by Eli Maor

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4

To the Limit, If It Exists

I saw, as one might see the transit of Venus, a quantity passing through infinity and changing its sign from plus to minus. I saw exactly how it happened … but it was after dinner and I let it go.—SIR WINSTON CHURCHILL, My Early Life (1930)

At first thought, the peculiar behavior of the expression (1 + 1/n)n for large values of n must seem puzzling indeed. Suppose we consider only the expression inside the parentheses, 1 + 1/n. As n increases, 1/n gets closer and closer to 0 and so 1 + 1/n gets closer and closer to 1, although it will always be greater than 1. Thus we might be tempted to conclude that for “really large” n (whatever “really large” means), 1 + 1/n, to every purpose and extent, may be replaced by 1. ...

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