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.

It deserves to be better known that there are simple cases where one can give closed forms for some rearrangements of alternating series. Here are a couple of interesting examples based on results of Schlömilch in 1873. Many further results can be found in classical textbooks on infinite series, e.g. those by Bromwich and Knopp.
Let $\rm\ H^m_n\ $ be the rearrangement of the alternating harmonic series $\rm\ 1 - 1/2 + 1/3 - 1/4 +\: \cdots\ $ obtained by taking consecutive groups of $\rm\:m\:$ positive terms and $\rm\:n\:$ negative terms. Then
$$\rm H^m_n\ =\ log\ 2 + \frac{1}2\ \lim_{k\to\infty}\ \int^{\:mk}_{nk}\frac{1}x\ dx\ =\ \log 2 + \frac{1}2 \log\frac{m}n $$
Similarly rearranging Lebniz's sum $\rm\ L\ =\ \pi/4\ =\ 1 - 1/3 + 1/5 - 1/7 +\: \cdots\ $ yields
$$\rm L^m_n\ =\ \frac{\pi}4 + \frac{1}2\ \lim_{k\to\infty}\ \int^{\:mk}_{nk}\frac{1}{2x-1}\ dx\ =\ \frac{\pi}4 + \frac{1}4 \log\frac{m}n $$
Thus as $\rm\:m\:$ varies we obtain infinitely many rearrangements with distinct sums. 
The proof of the general theorem underlying these results is quite simple - using nothing deeper than the integral test. See Beigel: Rearranging Terms in Alternating Series, Math. Mag. 1981.
