Let a1=√(2) and let an+1=√(2 an) for n ≥1. Show that (an) converges and find its limit. [Hint: S

step 1 Okay, so here we're given that a sub 1 is equal to the square root of 2, A sub n is equal to the

Let a1=√(2) and let an+1=√(2 an) for n ≥1. Show that (an)...

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Key Concepts

Fixed Point Analysis

Fixed point analysis involves finding a value that remains unchanged by a given function. When applied to a recursive sequence, one assumes that the limit exists and must satisfy the same relation as the terms of the sequence. Solving the resulting fixed point equation helps in determining the value of the limit for the convergent sequence.

Monotone Convergence Theorem

The monotone convergence theorem states that every bounded and monotonic (increasing or decreasing) sequence converges. This theorem is a fundamental result in real analysis and provides a clear criterion to conclude the convergence of sequences once monotonicity and boundedness are established.

Monotonic Sequence

A monotonic sequence is one that is either entirely non-decreasing or non-increasing. Proving that a sequence is monotonic, such as showing it is increasing (each term is greater than or equal to the one before), is a common step in establishing the convergence of recursively defined sequences.

Bounded Sequence

A bounded sequence is one that does not exceed a certain fixed upper and/or lower limit. Demonstrating that a sequence is bounded is crucial because it prevents the possibility of divergence by ensuring that the sequence’s terms remain within a finite range.

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