Find the sum of the infinite geometric series given by ∑_(k=1)^∞▒〖(2/9)^k.〗
To find the sum of an infinite geometric series, we can use the formula:
S = a / (1 - r),
where S represents the sum of the infinite series, a represents the first term, and r represents the common ratio between consecutive terms.
In this case, the first term (a) is (2/9)^1 = 2/9, and the common ratio (r) is 2/9.
Plugging these values into the formula, we get:
S = (2/9) / (1 - 2/9).
To simplify this expression, we need to find a common denominator for 1 and 2/9, which is 9.
S = (2/9) / (9/9 - 2/9).
Now, we can subtract the numerators: (9 - 2) = 7.
S = 2/9 / 7/9.
Dividing by a fraction is equivalent to multiplying by its reciprocal, so we can rewrite the expression as:
S = (2/9) * (9/7).
Simplifying further, we can cancel out the common factor of 9:
S = 2/7.
Therefore, the sum of the given infinite geometric series is 2/7.