Finding the limit of a sequence?

I'm a little confused with this example of finding the limit to a convergent sequence:

"A sequence is defined by the relation $a_{k+1} = -0.5 a_k +2$. Find L, the limit of $a_k$ as k tends to infinity."

The solution they have given is:

$Here, f(a_k) = -0.5a_k +2$

Now take $L=f(L)$, giving $L = -0.5L +2$

This rearranges to $L = \frac{4}{3}$

The point is that $\displaystyle \lim_{k \rightarrow \infty} a_k = \lim_{k \rightarrow \infty} a_{k+1}$.

So *if* the limit exists, then it must have the same value, say $L$, in both cases.

To show rigorously that it does indeed exist, you usually show (somehow) that the sequence $a_k$ is increasing and bounded above, or something along those lines.

Thank you. How would you tell if a sequence is convergent or divergent?

If you only have a sequence defined by a recurrence relation, you have to use an argument similar to the one that I gave above. If you have a formula for the terms of the sequence (e.g. $a_n= \frac{n}{n+1}$) you have to examine its behaviour carefully (which can be tricky) using a variety of techniques.

Are you an A level student or an undergraduate? If the former, then you won't have to worry about convergence issues much - you'll be told if the sequence converges or not, if necessary.

A level student revising for a Uni admissions test xD

Is there a way to find the general formula of a sequence given by its recurrence relation?

There are approaches that allow you to do this, but I don't recall the details too well, and I'm not sure if they work for all recurrence relations. Google is your friend here (generating functions is a good start).