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Generally speaking, it is incorrect to manipulate infinite series as if they were finite sums. For example, if zeroes are inserted into arbitrary positions of a divergent series, it is possible to arrive at results that are not self-consistent, let alone consistent with other methods. In particular, the step 4c = 0 + 4 + 0 + 8 + ⋯ is not justified by the additive identity law alone. For an extreme example, appending a single zero to the front of the series can lead to inconsistent results.[1] One way to remedy this situation, and to constrain the places where zeroes may be inserted, is to keep track of each term in the series by attaching a dependence on some function.[10] In the series 1 + 2 + 3 + 4 + ⋯, each term n is just a number. If the term n is promoted to a function n−s, where s is a complex variable, then one can ensure that only like terms are added. The resulting series may be manipulated in a more rigorous fashion, and the variable s can be set to −1 later. The implementation of this strategy is called zeta function regularization.

INTO JAPANESE

一般的に言えば、無限級数を有限和である場合の操作必要はありません。たとえば、ゼロは発散級数の任意の位置に挿入されている場合、ない自己無撞着ではおろか、他の方法で一貫性のある結果に到着することが可能です。特に、手順 4 c = 0 + 4 + 0 + 8 + ⋯ は、justifie ではありません。