(a) If {an} is convergent, show that limn →∞ an+1=limn →∞ an (b) A sequence {an} is defined

step 1 First part, let limit n tending to infinity nth term is l. It means that for every element to be

(a) If {an} is convergent, show that limn →∞ an+1=limn →∞ an...

(a) If $\left\{a_{n}\right\}$ is convergent, show that $$\lim _{n \rightarrow \infty} a_{n+1}=\lim _{n \rightarrow \infty} a_{n}$$ $\begin{array}{l}{\text { (b) A sequence }\left\{a_{n}\right\} \text { is defined by } a_{1}=1 \text { and }} \\ {a_{n+1}=1 /\left(1+a_{n}\right) \text { for } n \geqslant 1 . \text { Assuming that }\left\{a_{n}\right\} \text { is }} \\ {\text { convergent find its limit }}\end{array}$

(a) If $\left\{a_{n}\right\}$ is convergent, show that
$$\lim _{n \rightarrow \infty} a_{n+1}=\lim _{n \rightarrow \infty} a_{n}$$
$\begin{array}{l}{\text { (b) A sequence }\left\{a_{n}\right\} \text { is defined by } a_{1}=1 \text { and }} \\ {a_{n+1}=1 /\left(1+a_{n}\right) \text { for } n \geqslant 1 . \text { Assuming that }\left\{a_{n}\right\} \text { is }} \\ {\text { convergent find its limit }}\end{array}$

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Key Concepts

Fixed Point Equation

When a sequence is defined by a recurrence relation where the next term is a function of the current term, the limit of the sequence (if it exists) often satisfies a fixed point equation. This equation is obtained by setting the limit equal to its own image under the function, and solving it yields the candidate values for the limit of the sequence.

Convergence of Sequences

A sequence is said to be convergent if its terms approach a specific value, known as the limit, as the index goes to infinity. This concept is fundamental in analysis and implies that the distance between the sequence terms and the limit becomes arbitrarily small for all sufficiently large indices.

Shifted Sequence and Limit

For any convergent sequence, shifting the index by a fixed number does not affect the limit. That is, if a sequence converges to a limit L, then the sequence formed by a_{n+1} also converges to L. This property is important to simplify analysis in series and recurrence relations.

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