   Chapter 11.1, Problem 92E

Chapter
Section
Textbook Problem

(a) Show that if lim n → ∞ a 2 n = L and lim n → ∞ a 2 n + 1 = L , then { a n } is convergent and lim n → ∞ a n = L .(b) If a 1 = 1 and a n + 1 = 1 + 1 1 + a n find the first eight terms of the sequence { a n } . Then use part (a) to show that lim n → ∞ a n = 2 . This gives the continued fraction expansion 2 = 1 + 1 2 + 1 2 + ⋅ ⋅ ⋅

To determine

Part (a)

To show:

If limna2n=L and limna2n+1=L, then an is convergent and limnan=L.

Explanation

1) Concept:

Use definition 2 to show that an is convergent and limnan=L.

2) Definition 2:

A sequence an has the limit L, and we write limnan=L or anL as n.

If for every ε>0, there is a corresponding integer N such that if n>N, then an-L<ε

3) Given:

limna2n=L and limna2n+1=L

4) Calculation:

By definition 2, let ε>0.

Since limna2n=L, there exist N1

such that if n>N1

then a2n

To determine

Part (b)

To find:

(i) First eight terms of sequence an

(ii) Use part (a) show that limnan=2

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