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Helpppp confused


∑ 2^n
n=0

This equals to 1+2+4+8+16+...
(2-1)(1+2+4+8+16+...)
(2+4+8+16+32+...)-(1+2+4+8+16+32+...)
Numbers cancel out and we are left with -1.
I am very confused...
:(
Did I make a mistake??????
I am confused.

User Riik
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3.9k points

1 Answer

13 votes

Not an expertise on infinite sums but the most straightforward explanation is that infinity isn't a number.

Let's see if there are anything we missed:

Σ 2^n=1+2+4+8+16+...

n=0

We multiply (2-1) on both sides:

(2-1) Σ 2^n=(2-1)1+2+4+8+16+...

n=0

And we expand;

Σ 2^n=(2+4+8+16+32+...)-(1+2+4+8+16+...)

n=0

But now, imagine that the expression 1+2+4+8+16+... have the last term of 2^n, where n is infinity, then the expression of 2+4+8+16+32+... must have the last term of 2(2^n), then if we cancel out the term, we are still missing one more term to write:

Σ 2^n=-1+2(2^n)

n=0

If n is infinity, then 2^n must also be infinity. So technically, this goes back to infinity.

Although we set a finite term for both expressions, the further we list the terms, they will sooner or later approach infinity.

Yep, this shows how weird the infinity sign is.

User Happydude
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3.6k points