| 单词 | convergent series |
| 释义 | convergent series A series for which the sum of the first N terms tends to a finite limit as N tends to infinity. For example, in the geometric progression 1, r, r2,…,rN, where -1 < r < 1, the sum of the first N terms tends to a finite limit as N increases. This can be shown as follows; denote the sum of the first N terms by SN. Thus SN = 1 + r + r2 + r3 + …+ rN-1. Multiplying each term by r, rSN = r + r2 + …+ rN - 1 + rN. Subtracting the second series from the first, SN - rSN = 1 - rN. As N increases without limit, rN tends to zero, so (1 - r) SN tends to 1 and SN tends to 1/(1 - r). One might at first sight expect a series to be convergent if its individual terms tend to zero as N increases, but this is not correct. Consider for example the series 1, 1/2, [frac13],…[frac1N]; the first term is 1, the second 1/2. The next two terms sum to over 2(1/4) = 1/2 the next four sum to over 4 ([frac18]) = 1/2 and by taking successive groups of terms it is clearly possible to add amounts larger than 1/2 indefinitely, so the series is not convergent. |
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