I need some help!
Determine whether infinite geometric series has a finite sum. If so, find the limiting value.
112+56+28+....
To determine whether the given infinite geometric series has a finite sum, we need to check if the common ratio between consecutive terms is between -1 and 1. If it is, the series has a finite sum.
In this case, we can see that each term is half the value of the previous term. So the common ratio, r, is 1/2. Since the common ratio is between -1 and 1, the series has a finite sum.
To find the limiting value or the sum of the infinite geometric series, we can use the formula for the sum of an infinite geometric series:
S = a / (1 - r)
where S is the sum, a is the first term, and r is the common ratio.
In this case, the first term is 112 and the common ratio is 1/2. Substituting these values into the formula:
S = 112 / (1 - 1/2)
S = 112 / (1/2)
S = 112 * (2/1)
S = 224
So the sum of the infinite geometric series is 224.