Examples for Convergent Sequences

Example: Define x₀ = b and let x_n = x_{n - 1} / 2 for all n > 0. Then this sequence converges for any number b.

The proof is very easy using the theorem on monotone, bounded sequences:

b > 0: then the sequence is decreasing and bounded below by 0.
b < 0: then the sequence is increasing and bounded above by 0
b = 0: the sequence is constantly equal to zero

In either case the sequence converges. As to finding the actual limit, we proceed as follows: we already know that the limit exists. Call that limit L. Then we have:

lim x_n = L = lim x_{n + 1}

But then we have that

L = lim x_{n + 1} = lim x_n / 2 = 1/2 lim x_n = 1/2 L

so that we have the equation for the unknown limit L:

L = 1/2 L

Therefore, the limit must be zero.

This proof illustrates the advantage of knowing that a sequence converges. Based on that fact it was easy to determine the actual limit of this recursively defined sequence. On the other hand, it would be very difficult to try to establish convergence based on the original definition of a convergent sequence.

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