# 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 *absolute value theorem*.

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).