To prove that the sum:
1 / (1 - x) + 1 / (1 - x) + 1 / (1 - x)
equals 1, we can combine the fractions into a single fraction:
1 / (1 - x) + 1 / (1 - x) + 1 / (1 - x) = (1 + 1 + 1) / (1 - x)
Simplify the numerator:
= 3 / (1 - x)
Now, we know that for any value of x where |x| < 1, the sum of the infinite geometric series:
1 + x + x^2 + x^3 + ... = 1 / (1 - x)
Since our series is equivalent to the sum of three identical geometric series, we can substitute 1/3 for x in the formula for the sum of an infinite geometric series:
1 / (1 - 1/3) = 1 / (2/3) = 3/2
So, the sum of our series is 3/2. Therefore, it doesn't equal 1.
