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The infinite series whose terms are the natural numbers 1 + 2 + 3 + 4 + · · · is a divergent series. The ''n''th partial sum of the series is the triangular number :, which increases without bound as ''n'' goes to infinity. Because the sequence of partial sums fails to converge to a finite limit, the series does not have a sum. Although the series seems at first sight not to have any meaningful value at all, it can be manipulated to yield a number of mathematically interesting results, some of which have applications in other fields such as complex analysis, quantum field theory, and string theory. Many summation methods are used in mathematics to assign numerical values even to a divergent series. In particular, the methods of zeta function regularization and Ramanujan summation assign the series a value of −1/12, which is expressed by a famous formula: : In a monograph on moonshine theory, Terry Gannon calls this equation “one of the most remarkable formulae in science”. ==Partial sums== (詳細はPythagoreans as early as the sixth century B.C.E. Numbers of this form are called triangular numbers, because they can be arranged as an equilateral triangle. The infinite sequence of triangular numbers diverges to +∞, so by definition, the infinite series 1 + 2 + 3 + 4 + ⋯ also diverges to +∞. The divergence is a simple consequence of the form of the series: the terms do not approach zero, so the series diverges by the term test. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「1 + 2 + 3 + 4 + ⋯」の詳細全文を読む スポンサード リンク
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