Math Problem 6: Sum of the Infinite Series 1 - 1 + 1 ...
Consider the infinite sum 1 - 1 + 1 - 1 + 1 .... Let us set it equal to x, so x = 1 - 1 + 1 - 1 ....
First, rewrite this equation as follows: x = 1 - (1 - 1 + 1 - 1 ...), notice that the infinite sum in the parenthesis is the same as x, so we can substitute back in for x and get x = 1 - x, so x = 1/2.
Now, consider the original infinite sum again; this time, put every two terms in parenthesis: x = (1 - 1) + (1 - 1) ..., simplify each parenthesised term: x = 0 + 0 ..., so x = 0.
How can x be both 1/2 and 0? Which answer is correct?
Neither answer is correct. There is something wrong with both answers.
There is no correct solution for the sum of that infinite series, because that series does not converge. Any method of obtaining an answer must therefore be wrong.
The problem with the first answer is that x is used as though it were a definite number, when in fact it is not a number at all.
The problem with the second answer is that parenthesis are used, which implies associativity. The principle of associativity cannot be applied in this situation, since we are not dealing with definite numbers.