Question #86545
2 Answers
The limit is
It makes sense because both
In fact, you can prove that
using
and the discrete version of the squeeze theorem (aka sandwich or pinch theorem).
For
To determine this limit, the best way to do so is using the idea of the squeeze theorem. Similar in Calculus I for limits of functions, this idea states that as long as
This means that as the two sequences converge to some number or diverge, the sequence in the middle (the original sequence) must also do the same since it is between the two as they "squeeze" to the same final result. Here is a good example online:
http://supportcentre.maths.nuim.ie/documents/uploads/user/SqueezeTheoremforSequences.pdf
For alternating sequences like
If you take the absolute value of a sequence
Let's begin by taking the absolute value of the sequence we are solving for:
We can assume
So,
Since this is always the case for alternating sequences, you can just show the absolute sequence approaching to zero and state what theorem you used (squeeze theorem or absolute value theorem).