Please explain how Conditionally Convergent can be valid? I understand the basic idea of Conditionally Convergent (some infinitely long series can be made to converge to any value by reordering the series). I just do not understand how this could possibly be true. I think it defies common sense and seems like a clear violation of the Commutative Law.

When it comes to infinity, most common sense goes out the window. Cantor had a hard time convincing some of his peers...
For me what was surprising is that absolutely convergent series can be reordered in any fashion and still give the same result!
When begining to learn, one naturally considers the series $\displaystyle 1 - 1 + 1 - 1 \dots$
This can be written as
$\displaystyle  1 - (1-1) - (1-1) \dots = 1$
or
$\displaystyle (1-1) + (1-1) + \dots = 0$
Now it should not be a (big) surprise if we could reorder some series differently to give multiple possible values. From there it should not be a big surprise if there were some series which could give any possible value. 
The proof of Riemann Rearrangement Theorem might help reduce the surprise :-)
Hope that helps.
