Questions About Infinite Sums

I was reading the Wikipedia article about the sum 1+2+3+4+..., and I saw this explanation:

c = 1+2+3+4+5+6+...

4c = _4__+8__+12+...

-3c = 1-2+3-4+5-6+...

link: http://ift.tt/1duoR32

My question, as one who hasn't worked with infinite sums:

Why are you allowed to shift the numbers when adding/subtracting/manipulating infinite series. For instance:

b = 1+1+1+...

b = __1+1+...

thus b-b = 0 = 1

If shifting numbers is allowed, why can something like that be accounted for? Is it a dividing by zero, "dont touch that" kind of thing or is shifting series while manipulating them only allowed for certain series?

Also on Wikipedia (link: http://ift.tt/1jJgGoG), I saw that the sum of 1+1+1+... = -1/2. If you add an infinite number of 1+1+1+... together after shifting them, you can make the original 1+2+3+4+...

Here is what I am saying:

b = 1+1+1+1+1+...

b = __1+1+1+1+...

b = ____1+1+1+...

and so on...

So if 1+1+1+... = b, b = -1/2, b+b+b+... = 1+2+3+4+... and 1+2+3+4+... = -1/12 does (-1/2)+(-1/2)+(-1/2)+... = -1/12?

Answers to those questions would be tremendously appreciated, as well as any critiques of my misunderstanding of this subject. Thank you for your time.

Bonus question: Has anyone figured out how an infinite sum of positive numbers equals a negative number? I'm not asking for proofs of the sum, just an explanation of this weird result.

p.s. Sorry for the underscores, I had trouble with the formatting.