Suppose that s1>s2>0 and let sn+1=(1)/(2)(sn+sn-1) for n ≥2. Show that (sn) converges. [Hint: Sh

step 1 There is we have a sub n is 1. Then a sub 2 would be 1 half times 1. So that's 1 half. Then a su

Suppose that s1>s2>0 and let sn+1=(1)/(2)(sn+sn-1) for n ≥2....

Suppose that $s_1>s_2>0$ and let $s_{n+1}=\frac{1}{2}\left(s_n+s_{n-1}\right)$ for $n \geq 2$. Show that $\left(s_n\right)$ converges. [Hint: Show that $\left(s_{2 n-1}\right)$ decreases and $\left(s_{2 n}\right)$ increases.]

Key Concepts

Bounded Sequences

Boundedness ensures that the sequence's terms do not diverge to infinity or minus infinity. When a sequence is shown to be bounded and monotonic, it satisfies conditions that lead to the convergence of the sequence, which is a central concept in real analysis.

Convergence of Sequences

Convergence refers to the behavior of a sequence as its index increases without bound, specifically approaching a finite limit. Demonstrating convergence commonly involves showing that successive differences between terms become arbitrarily small, or that a sequence exhibits both monotonicity and boundedness, thereby guaranteeing its convergence by classical theorems.

Monotonic Sequences

A monotonic sequence is one that consistently increases or decreases. Proving that a sequence or its subsequences are monotonic is crucial because monotonic sequences that are bounded are guaranteed to converge, according to the Monotone Convergence Theorem.

Recurrence Relations

Recurrence relations define sequences by expressing each term as a function of preceding terms. Analyzing these relations often involves finding fixed points or limits and allows the use of induction or iterative techniques to establish properties such as monotonicity and boundedness, leading to conclusions about convergence.

Subsequence Analysis

Subsequence analysis involves breaking the original sequence into parts (for example, terms with even or odd indices) and examining their individual behaviors. If these subsequences are monotonic and converge to the same limit, it can be inferred that the entire sequence converges, which is a powerful method to tackle problems with alternating behaviors.

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