Nested Sigma Noation

Just saw this in that other Rubiks Cube thread, and I have no idea how to do this....

Simplified version from what they were doing:


How do you find the sum of this? Or prove that it converges / diverges? What happens when you add a third? A fourth...?

There are some neat tricks here, which are handy whenever dealing with summations.
- As long as multiplications are only constants (relative to the summation), we can put it out of the summation, or in in the summation.
- If there are two or more summation signs "in a row", any permutation of their order will be equal to the original summation.
- You can split summations in two or more, and merge two or more summations as long as they equal the same sum.

So, even though people have shown that the series is clearly divergent, let me show you another way of showing that the series is divergent. Using the middle trick I mentioned, we can see that:



By splitting it up:



And we know that



So, clearly, this must diverge.

Also, whether individual terms converge or not does not matter: If the individual terms does in fact converge, express them as their value and not their sum. It should most likely help out. Ex. of individual converging series:

 Spoiler

The original series is only two replacements from converging though, as



converge. It's interesting to note that the series equal to 1 - you should try to prove it by using the examples mentioned earlier!

Noah