Question

Am I right about the answer you get when you add up to Infinity? Math Experts Only, Anyone else is Welcome too.

Answer 1

\[1+2+4+8+16... = \infty \]
Fact
\[1 \times x = x\]
Fact
\[(1)(1+2+4+8+16...) = 1+2+4+8+16...\]
Fact
\[2-1 = 1\]
Fact
\[(2-1)(1+2+4+8+16...) = 1+2+4+8+16...\]

ALSO

\[(2-1)(1+2+4+8+16...) = 2+4+8+32...-1-2-4-8-16-32...\]
Almost everything on the right side cancels out, guess what's left behind?
\[= -1\]
\[1+2+4+8+16... = -1\]
\[\infty = -1\]
How about that?

Answer 2

Correction:
\[(2-1)(1+2+4+8+16...) = 2+4+8+16+32...-1-2-4-8-16-32...\]

Answer 3

I may be wrong, but your sum would be equivalent to \[\sum_{n=0}^{\infty}2^{n}\] A rearrangement of a series would only converge to the same value if it were absolutely convergent, but this series is divergent. Thus I'm sure you could do a manipulation and get any result you want.

Answer 4

Fair enough, but this side of Maths is still interesting though, it isn't so 'Perfect'? If that's how someone would put it.

Answer 5

Of course, a lot of this stuff is fascinating in my eyes. It's what makes it so fun to study :)

Answer 6

I'll just leave this Question open for a bit and Bump it for others to see.
Thanks!

Answer 7

If the series is not absolutely convergent then the addition of infinite sums is not commutative.

You might want to take a look at https://en.wikipedia.org/wiki/Alternating_series#Rearrangements

Here we are talking about creating a convergent series from a divergent one. Since the original series is not absolutely convergent, then commutation is not allowed.