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Infinite Sequences and Series

Infinite Sequences and Series In this chapter we shall study the theory of infinite sequences and series, and investigate their convergence. Examples ：. 若 A={-1,-2,-3,-4,…}, 則 A 是有上界的集合，且 -1,0,1, 皆是 A 的一個上界 ， 其實大於或等於 -1 的實數都是 A 的上界 。

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Infinite Sequences and Series

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  1. Infinite Sequences and Series In this chapter we shall study the theory of infinite sequences and series, and investigate their convergence.

  2. Examples： • 若A={-1,-2,-3,-4,…}, 則A是有上界的集合，且-1,0,1,皆是 A的一個上界，其實大於或等於-1的實數都是A的上界。 • 若A={1,2,3,4,5,…},則A是有下界的集合，且0,-1,-2,皆是 A的一個下界，其實小於或等於1的實數都是A的下界。 • 若A={-3,-2,1,0,1,2,3,4}, 則A有一個上界4及有一個下界-3 • 故A是一個有界集合。

  3. Definition: • 令A是有上界的集合，若 是A的一個上界且 小於或等於A的其他上界，則 稱為A的最小上界，記為lub(A) 或sup(A) 即 lub(A)=sup(A)= • 令A是有下界的集合，若g是A的一個下界且g大於或等於A的其他下界，則g稱為A的最大下界，記為glb(A) 或inf(A) 即 glb(A)=inf(A)=g. • 注意1.若A是有上界的集合，則sup(A)存在。 • 2.若A是有下界的集合，則 inf(A)存在。 • 3.若A是有界集合，則sup(A)及inf(A)存在。 • Example: • 若A={x | x<0}, 則lab(A)=sup(A)=0, 但sup(A) A 。 • 若A={1/n | n=1,2,3,…}, 則lub(A)=1, glb(A)=0, 但是 。

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