Define a recursive sequence by a1 = 2 and an+1 = 1 − a2n. Either prove that (an) converges or prove that it diverges.
Define a recursive sequence by a1 = 2 and an+1 = 1 − a2n. Either prove that (an) converges or prove that it diverges.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Define a recursive sequence by a1 = 2 and an+1 = 1 − a2n. Either prove that (an) converges or prove that it diverges.

Transcribed Image Text:**Problem 2: Recursive Sequence Analysis**
Define a recursive sequence by \( a_1 = \frac{1}{2} \) and \( a_{n+1} = 1 - a_n^2 \). Either prove that \( (a_n) \) converges or prove that it diverges.
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### Explanation
This problem involves a recursive sequence, which means each term is defined in terms of the previous term. To address the problem, one must determine whether this sequence approaches a specific value (converges) or continues indefinitely without settling on a particular value (diverges).
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