Find the sum to infinity of the G.P. 1-1/2+1/4-....
In order to find the sum to infinity of the given geometric progression, we need to determine if the series converges to a finite value or if it diverges to infinity.
The given geometric progression is:
1 - 1/2 + 1/4 - 1/8 + ...
We can see that each subsequent term is half the value of the previous term, indicating that the common ratio (r) is 1/2.
Now, according to the formula for the sum of an infinite geometric series, if the absolute value of the common ratio (|r|) is less than 1, the series converges to a finite value. If |r| is greater than or equal to 1, the series diverges.
In this case, we have |r| = |1/2| = 1/2, which is less than 1. Therefore, we can conclude that the given geometric progression converges to a finite value.
To find the sum to infinity (S∞) of this geometric progression, we can use the formula:
S∞ = a / (1 - r)
where a is the first term and r is the common ratio.
In this case, a = 1 and r = 1/2. Substituting these values into the formula, we get:
S∞ = 1 / (1 - 1/2)
= 1 / (1/2)
= 2
Therefore, the sum to infinity of the geometric progression 1 - 1/2 + 1/4 - ... is 2.