   Chapter 11.1, Problem 70E

Chapter
Section
Textbook Problem

(a) If { a n } is convergent, show that lim n → ∞ a n + 1 = lim n → ∞ a n (b) A sequence { a n } is defined by a 1 = 1 and a n + 1 = 1 / ( 1 + a n ) for n ≥ 1 . Assuming that { a n } is convergent, find its limit.

To determine

(a)

To show:

limnan+1=limnan

Explanation

1) Concept:

A sequence an has the limit L if for every >0  there is a corresponding integer N such that if n>N then an-L<

2) Definition:

Convergent: If limnan exists then the sequence is called convergent

Divergent: If limnan does not exist then the sequence is called divergent

3) Given:

an isa convergent sequence

4) Calculation:

It is given that  an is a convergent sequence

By definition of a convergent sequence,

limnan exists

Let us consider the value of limit as L

limnan=L

By using the definition of limit of sequence,

For every >0  there is a corresponding in

To determine

(b)

To find:

limnan

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