Show that the sequence defined by a1=2 an+1=(1)/(3-an) satisfies 0<an ⩽2 and is decreasing. De

Show that the sequence defined by a1=2 an+1=(1)/(3-an)...

Key Concepts

Recursive Sequence

A recursive sequence is one where each term is defined in relation to previous term(s) using a recurrence relation. Such sequences allow the investigation of their long?term behaviour by studying the function that maps one term to the next.

Bounded Sequence

A bounded sequence is one that remains within fixed upper and lower limits. Establishing that a sequence is bounded is a crucial step in proving convergence because it prevents the terms from diverging to infinity or negative infinity.

Monotonic Sequence

A monotonic sequence is one that is either entirely non-increasing or non-decreasing. Demonstrating that a sequence is monotonic helps in proving convergence, especially when combined with boundedness.

Monotone Convergence Theorem

The Monotone Convergence Theorem states that every sequence which is both monotonic (non-decreasing or non-increasing) and bounded must converge. This theorem is a central result in real analysis for establishing the convergence of such sequences.

Fixed Point Analysis

Fixed point analysis involves finding a value L such that when applied to the recursive function, it remains unchanged (L = f(L)). When a recursive sequence converges, its limit is a fixed point of the function that defines the recurrence, and solving the fixed point equation helps determine this limit.

Transcript

00:01 Given that a1 equal to 1 and a n plus 1 equal to this, we first want to show that a .n is within the interval 0 and 2 for all n in n, which is a sign natural number.

00:36 So n equal to 1 is true.

00:46 If it's true, then we have zero a1 to be equal to 0 is less than a1, which is equal to 2, less than or greater than 2.

01:00 Assume, assume that for some n a natural number, n in natural number, any natural number you have the interval of a n 0 to 2 then this implies that we have 0 than a n less than or equal to 2 so you have 0 greater than minus a n greater than or equal to then you have three greater than 3 minus a .n, which is greater than or equal to 1 divided by 3.

02:02 And this is less than.

02:09 Okay, so then we have, so this implies that this implies we have, we have one divided by three, less than one, divided by 3 minus a n less than or equal to 1 which implies that 1 divided by 3 is less than a n plus 1 a and plus 1 less than or greater than 1 so this implies that 0 is less than a n plus 1 less than to two.

03:09 So by mathematical induction, so this by mathematical, mathematical induction, induction, we have, we get zero to be less than a n, less than or equal to two is true for all n in natural number.

03:44 So this implies that the sequence the sequence the sequence the sequence is bounded so if we show if we show that if we show that it is it is monotonic to then we then we will know the sequence converges the sequence converges the sequence converges so we have a two say if we pick a to be equal to one divided by three minus a one this is equal to 1 divided by 3 minus 2 which is equal to 1.

05:15 So this implies that a2 is less than a1.

05:21 So assume that for some n, a natural number, a .n plus 1, a .n plus 1 is less than a .n.

05:45 So then it implies that a .n.

05:51 Plus 1 is less than a .n.

05:56 So negative a .n.

05:58 Plus 1 will be greater than minus a .n.

06:03 So 3 minus a .n...

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