Limit sup and limit inf.

Limit sup and Limit order to make us understand the information more on approaches of a given realsequence an n 1 , we give two definitions, thier names are upper Limit and lower Limit . Itis fundamental but important tools in of Limit sup and Limit infDefinition Given a real sequence an n 1 ,wedefinebn sup am:m n andcn inf am:m n .Example 1 1 n n 1 0,2,0,2,.. ,sowehavebn 2andcn 0 for 1 nn n 1 1,2, 3,4,.. ,sowehavebn andcn for n n 1 1, 2, 3,.. ,sowehavebn nandcn for Given a real sequence an n 1 , and thus definebnandcnas the same ,andcn n If there is a positive integerpsuch thatbp , thenbn n there is a positive integerqsuch thatcq , thencn n bn is decreasing and cn is property 3, we can give definitions on the upper Limit and the lower Limit of a givensequence as Given a real sequence an and letbnandcnas the same as before.

Relations with convergence and divergence for upper (lower) limit Theorem Let an be a real sequence, then an converges if, and only if, the upper limit and the lower limit are real with lim n supan lim n infan lim n an. Theorem Let an be a real sequence, then we have

Loading..

Tags:

  Convergence, Divergence, Convergence and divergence

Information

Related search queries