Problem with this weird series So, I came across a problem in 'Complex Variables by Schaum'

Given, $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}...$converges to S


Prove that the rearranged series:
$$1+\frac{1}{3}-\frac{1}{2}+\frac{1}{5}+\frac{1}{7}-\frac{1}{4}+\frac{1}{9}+\frac{1}{11}-\frac{1}{6}+...=\frac{3}{2}S$$
(Hint: Take half of first series and write it as $0+\frac{1}{2}+0-\frac{1}{4}$ and add term by term to the first series.

Now the hint pretty much does it. When you take half of S and add term by term to S you do get the asked series and its proved.
My issue:
The asked series can itself be re-written as:
$$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}....=S$$
The asked series is basically the same series as the first one and yet their summations are different. How could that be possible??
What is the noob mistake I am doing?
 A: 
Bernhard Riemann proved that a conditionally convergent series may be
rearranged to converge to any value at all, including ∞ or −∞; see
Riemann series theorem.

https://en.wikipedia.org/wiki/Conditional_convergence
https://en.wikipedia.org/wiki/Riemann_series_theorem
A: The plain fact is:  Infinity breaks everything.
We're used to addition being commutative and associative.  But take the series
$$1 - 1 + 1 - 1 + \cdots$$
Associate one way:
$$(1 - 1) + (1 - 1) + \cdots = 0 + 0 + \cdots = 0.$$
Associate another way:
$$1+ (-1+1) + (-1+1) +\cdots = 1+ 0 + 0 +\cdots = 1.$$
So associativity fails to work on infinite sums.  Your example shows that commutativity fails to work on infinite sums.  In Calculus, you learn about "absolute convergence".  And the whole point of absolute convergence is that it restores associativity and commutativity (and many other things) for certain infinite sums.
A: Remember that in an important sense, an infinite series isn't actually a sum, so you can't necessarily expect it to obey the laws applicable to sums. An infinite series is defined as a limit of partial sums. It's reasonable that if you rearrange the terms that result in those partial sums, you'll get different partial sums that result in a different limit. Indeed, from this perspective, the more surprising result is that there are some conditions (such as absolute convergence) that let you get away with rearranging terms.
