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词汇 Limit of a sequence
分类 英语词汇 英语翻译词典
释义

Limit of a sequence

中文百科

极限 (数列)

极限，即为一个数列\{a_n\}，使得\lim_{n \to \infty}a_n=L，其中L为一确定的常数，亦即数列\{a_n\}随着n的增加而趋近于L。

设\{ x_n \}, x_n \in \mathrm R,n=1,2,\ldots,x_0 \in \mathrm R，

对于任意的正实数\epsilon，存在自然数N，使得当n>N时，有 |x_n-x_0 | < \epsilon，

用符号来表示即 \forall \epsilon >0 ,\exists N \in \mathbb N,\forall n>N,|x_n-x_0 | < \epsilon

则称数列\{ x_n \}收敛于x_0，记作\lim _{n \to \infty} x_n=x_0。

英语百科

Limit of a sequence 极限 (数列)

The sequence given by the perimeters of regular n-sided polygons that circumscribe the unit circle has a limit equal to the perimeter of the circle, i.e. . The corresponding sequence for inscribed polygons has the same limit.
The plot of a convergent sequence an is shown in blue. Visually we can see that the sequence is converging to the limit 0 as n increases.
The plot of a Cauchy sequence (xn), shown in blue, as xn versus n. Visually, we see that the sequence appears to be converging to a limit point as the terms in the sequence become closer together as n increases. In the real numbers every Cauchy sequence converges to some limit.

In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to". If such a limit exists, the sequence is called convergent. A sequence which does not converge is said to be divergent. The limit of a sequence is said to be the fundamental notion on which the whole of analysis ultimately rests.
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