Why does rearranging terms in an infinite series make two series convergent and divergent for the same values. Shouldn't the sum be the same. After all we are not adding new terms. Can someone explain the concept behind it?
For example from wiki (https://en.wikipedia.org/wiki/Riemann_zeta_function):
$\zeta(s) = \sum n^{-s}, \Re(s) > 1$
An extension of the area of convergence can be obtained by rearranging the original series.The series
$\zeta(s) = 1/(s-1)\sum_{n=1}^\infty (\frac {n} {{(n+1)}^s} - \frac {n-s} {n^s})$ converges for $\Re(s) > 0$
I know it has to do something with conditional convergence. But intuitively I am struggling to see why it works since the set of values in both series is same/unique the sum must also be unique. Or are we doing sum jugglery here relative to some predefined definition of what it means to be convergent or divergent? Sum of the values of any set must be unique so why does ordering matter?