Hi forum,
I saw this trollmath on youtube. i dont claim its correct, but i cant understand where the mistake is. help
Consider a geometric series as follows:
[1]
[2]
[2]-[1]
Clearly the last line is false, but where is the mistake?
That is not exactly correct. You can subtract infinite sums if they converge. For example, 1+ 1/2+ 1/4+ 1/8+ ... (adding 1 over powers of 2) is a "geometric series" that converges to 1/(1- 1/2)= 2. Similarly, 1+ 1/3+ 1/9+ 1/27+ ... (adding 1 over powers of 3) is a geometric series that converges to 1/(1- 1/3)= 3/2. (1+ 1/2+ 1/4+ 1/9+...)- (1+ 1/3+ 1/9+ 1/27+ ...)= (1/2- 1/3)+ (1/4- 1/9)+ (1/8- 1/27)+ ...= 1/6+ 5/36+ 19/216+ ... converges to 2- 3/2= 1/2.
The problem here is that the series does not converge.
Hello, SpringFan25!
Consider a geometric series as follows:
. .
Clearly the last line is false, but where is the mistake?
The mistake is in thinking that we can "do" arithmetic with Infinity.
Clearly: .
Then: .
Hence: . , an indeterminate form.
That is, could be any value.
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Here is a classic "paradox".
Method 1
Method 2
Method 3
is a geometric series
. . with first term and common ratio
Hence: .
Therefore: .
We CAN get the right answer if we are careful
Subtract vertically to get S
Just because we have the difference of two infinite divergent sums doesn't necessarily put the right answer out of reach , like Soroban said infinity minus infinity could be anything... that includes the right answer. Finding a procedure that produces the right answer may be difficult though.