Gottfried Leibniz, in a letter to Johann Bernoulli, observed that an alternating series whose terms monotonically decrease to zero in absolute value is convergent. In a letter of January 10, 1714, he gave an incorrect proof. Examination of the proof indicates that it is essentially the one used today, except that Leibniz failed to say anything about the completeness of the real numbers.
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Alternating Series and Leibniz's Test