Rearrangements of a conditionally convergent series In the case of conditional convergence of a series, why do rearrangements affect the value of the series?
 A: consider the series $\sum_{i=1}^\infty (-1)^{i+1}\cdot \frac1i$. It is (conditionally) convergent. We now construct a divergent rearrangement. Since $\sum_{i=1}^\infty \frac 1{2i+1}$ diverges, we can choose an increasing sequence $(N_k)$ with $N_0 = 0$ in $\mathbb N$ s.th. $\sum_{i=N_k+1}^{N_{k+1}} \frac 1{2i+1} \ge 1$ for all $k\in \mathbb N$. Now 
$$
  \sum_{k=1}^\infty \left(\sum_{i=N_k+1} \frac 1{2i+1} - \frac 1{2i}\right)
$$ is a divergent rearrangement of the given series. I hope, I understood your question correctly,
HTH, Yours, AB, martini.
A: The intuitive answer is that in a conditionally convergent series the positive terms sum to $+\infty$ while the negative terms sum to $-\infty$.  We know $\infty-\infty$ is not well defined.  If we sum up the positive terms faster than the negative ones we increase the value of the sum.  The link J. M. gave is a good one.
A: This is called Riemann's rearrangement theorem. A better description than I could possibly type up here is given at wikipedia. It has detailed examples and a full proof.
