3.1. Sequences

Example 3.1.3(c):

The sequence converges to zero.

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{ n / 2n } = { 0, 1/2, 1/2, 3/8, 1/4, 5/32, ...}. It is not clear, but it seems as if the terms get smaller and smaller. Indeed this is the case, and we will prove it:

First, we can use induction to show that

n2 2n
for n > 3. But then we have that
n2 / 2n 1
or equivalently
n / 2n 1/n
for n > 3. But now you should be able to finish the proof yourself. As a hint, for a given , choose
N = max{3, 1/}
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