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单词 ExampleOfRatioTest
释义

example of ratio test


Consider the sequence given by an=xn (geometric progression) where |x|<1.Then the series

j=0an

converges.  To see this, we can use the ratio testMathworldPlanetmath. We need to consider the sequence |an+1/an|. But for any  n0  we have (when  x0)

|an+1an|=|xn+1xn|=|x|<1,

and therefore the series converges. The ratio test and the previous argument shows that the geometric seriesMathworldPlanetmath diverges for  |x|>1.

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更新时间:2025/5/4 5:16:48