An example of a series that diverges and exhibits no pattern?

[Permanence]

Hi I was hoping someone could provide me with a couple examples of series that exhibit no pattern. In my course we only cover divergence that involves the limit tending to infinity or oscillation. My textbook informs me that in some rare cases no pattern may be exhibited, but doesn't list any examples. I'm drawing a blank on brainstorming.

Thanks!

PS
Also are there any specific characteristics or tests that would clue me into knowing that it exhibits no patterns.

 

[jbunniii]

How about if you define $a_n$ to be the $n$'th decimal digit of $\pi$?

Then this series diverges:
$$\sum_{n=1}^{\infty}a_n$$
because the terms do not converge to zero, and this series converges
$$\sum_{n=1}^{\infty}\frac{a_n}{n^2}$$
by comparison with $\sum 10/n^2$.

 

[Permanence]

Gah I feel so stupid. I've seen that specific example before, but I never really thought about it that way. In our class we've always defined a(sub)n to be a particular function that we've seen before.

Thank you Jbunniii for the swift response!

 

[phyzguy]

How about:
$$\sum_{i=1}^\infty \frac{1}{p_n}$$
where p_n is the nth prime number? I think this also diverges.

 

[JohnWisp]

Gah I feel so stupid. I've seen that specific example before, but I never really thought about it that way. In our class we've always defined a(sub)n to be a particular function that we've seen before.

Thank you Jbunniii for the swift response!

For education a Liouville number is the universal example of such a series.

http://en.wikipedia.org/wiki/Liouville_number