Infinite Series (and Sequences) (cont.) (see first post) (parentheses)

When nobody knew anything about my blog except for my friend and some German Menschen (some of my first internet traffic came via Germany, strangely enough), I had a post about infinite serieses*. I found something interesting that my math teacher recommended I put in here, and as that topic was infinite sequences, I decided to put it in here.

Here is one of the problems:
Let's make a house of cards!

Write a formula for the sequence representing the number of cards at each stage.
First method is to make a chart.
Stage-     1      2      3      4...
Cards-    2      7     15    26...
                 +5    +8   +11
Each stage increases by 3 more than the last increase. Now I'm sure there's an algebraic way to solve it from there, but my method is much more interesting.
Look at the direction that the cards face-- I marked them with lines. What's the sequence that the directions make?
Stage-   1    2    3    4
Dir 1-   1    3    6    10
Dir 2-   1    3    6    10
Dir 3-   0    1    3     6
We know those sequences! Those are triangular numbers, denoted by (y=n(n+1)/2). Because one is offset to the left, we modify the formula a bit to (y=n(n-1)/2). By definition, we can add these three formulas together to get:
y=n(3n+1)/2


Another thing we thought about in class were divergent sequences with really weird sums. For example, 1+2+3+4... is equal to -1/12, while 1+2+4+8... is equal to -1. Here's why:
For the first, imagine three sequences:
S1=1-1+1-1+1-1+1...
S2=1-2+3-4+5-6...
S=1+2+3+4+5...
Unfortunately, S1 orbits on the period of 1 and 0. So we have to use other methods. Watch this:
S1 + S1 =   1 - 1 + 1 - 1 + 1 - 1 ....
                      + 1 - 1 + 1 - 1 + 1
              =   1 + 0 + 0 + 0 + 0 + 0... = 1
So two times S1 is one, so S1 is 0.5.
And then, there's S2.
S2 + S2 =   1 - 2 + 3 - 4 + 5 - 6 + 7 ...
                      + 1 - 2 + 3 - 4 + 5 - 6 ...
              =   1 - 1 + 1 - 1 + 1 - 1 ...
Thereby establishing that S2 is 0.25.
Then watch this!
S - S2 =   1 + 2 + 3 + 4 + 5 + 6 + 7 ...
               -1 + 2 - 3  + 4 -  5 + 6 -  7
           = 4 + 8 + 12 + 16 + 20 + 24 + 28 ...
which is 4 times S. We end with:
S - 0.25 = 4S
S = -1/12
Wow! Amazing! Literally impossible!
There's probably some reason that it's not true, but who cares? It's just time to marvel at the effects of calculus on algebra!

The next one on the list (that my friend learned from a high school teacher) is
S = 1 + 2 + 4 + 8 + 16 + 32 + 64
And so we use the same method, in theory.
S - S = 1 + 2 + 4 + 8 + 16 + 32 + 64 ...
                - 1 -  2  - 4  -  8  -  16  - 32..
         = 1 + 1 + 2 + 4 + 8 + 16 + 32...
And so
S - S = S + 1
S = -1

Another impossible sum! If any people disagree they will be ridiculed forever! I'm kidding- just give me a reason that I understand.

Stay coolio,
John

PS.: This post took FOREVER