A sequence is called convergent if there is a real number that is the limit of the sequence.

Convergent

sequences

Divergent

sequences

It has

a limit

It has

no limit

If there is no such number, then the sequence is divergent.

But there are degrees of divergence.

A sequence is divergent if it tends to infinity,

but it is also divergent if it doesn’t tend to anywhere at all.

Sequences that tend to nowhere are always oscillating sequences.

The simplest example of an oscillating sequence is the sequence.

But before we start to think that all oscillating sequences are divergent,

well, here comes another one.

So, just because a sequence bounces around, it isn’t necessarily divergent.

The size of those jumps is also important.

Here are three examples of the possible behaviors:

if n is even

if n is odd

And now let’s see some funny cases.

Well, here in the formula the 1 is up front, so let’s swap it.

It doesn’t pose a problem for addition.

For subtraction...

it is also OK, if we don’t mess it up.

And Voila! We got another oscillating sequence.

Now, appears both in the numerator and the denominator,

so this is not an oscillating sequence.

And now let’s see some sequences with radicals.

Here comes another one.

Again, we identify which member of the denominator has the highest exponent.

And now let’s see something really interesting.

Well, there is nothing exciting yet.

The thrill comes when we replace the + sign...

with the sign.

is also , but

In such cases we need some magic.

From here, it is the usual drill.

And then we have this:

And one more:

If we have addition here, then we are done.

But if it is subtraction, then we need some hocus-pocus again.