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

1. Apr 9, 2013

### 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.

2. Apr 9, 2013

### 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 $\sum 10/n^2$.

3. Apr 9, 2013

### 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!

4. Apr 9, 2013

### phyzguy

$$\sum_{i=1}^\infty \frac{1}{p_n}$$