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## Main Question or Discussion Point

I wasnt sure which math forum to ask this in so im putting it here, apologies if its supposed to be elsewhere.

Anyway, my question has to do with an infinite sum.

Suppose we have a series:

1+2+4+8+16+32+...

each term in the series is double the previous term starting with 1

if we multiply this by the number 1 it will remain the same

1+2+4+8+16+32+... = 1(1+2+4+8+16+32+...)

also (2-1) = 1

so we can say

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

if we expand the RHS we get

1+2+4+8+16+32+... = (2+4+8+16+32+64+...) - (1+2+4+8+16+32+...)

1+2+4+8+16+32+... = (2+4+8+16+32+64+...) + (-1-2-4-8-16-32-...)

the first part of the RHS has a set of all positive even numbers and the 2nd part of the set has a set of all negative even numbers

all the even numbers will subtract to 0 and i'm left with

1+2+4+8+16+32+... = -1

what have I done wrong? I can't figure what and where the error is.

Anyway, my question has to do with an infinite sum.

Suppose we have a series:

1+2+4+8+16+32+...

each term in the series is double the previous term starting with 1

if we multiply this by the number 1 it will remain the same

1+2+4+8+16+32+... = 1(1+2+4+8+16+32+...)

also (2-1) = 1

so we can say

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

if we expand the RHS we get

1+2+4+8+16+32+... = (2+4+8+16+32+64+...) - (1+2+4+8+16+32+...)

1+2+4+8+16+32+... = (2+4+8+16+32+64+...) + (-1-2-4-8-16-32-...)

the first part of the RHS has a set of all positive even numbers and the 2nd part of the set has a set of all negative even numbers

all the even numbers will subtract to 0 and i'm left with

1+2+4+8+16+32+... = -1

what have I done wrong? I can't figure what and where the error is.