What is the sum of the infinite geometric series?
\[ \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \ldots \]
Answer (3)
1/2 + 1/4 + 1/8 + 1/16 + ... = n = 1 ∑ ∞ ( 2 1 ) n = 1 − 2 1 2 1 = 1
We can see that consecutive fractions are made from 1/2 to consecutive powers. Because we begin with 1/2, n=1 We will infinitely add fractions , hence Lemniscate sign.
In other words you cannot find a number which needs to be added to the geometric series to get "1", therefore the answer is 1. I remember the teacher explaining it this way :)
The sum of the given geometric series is, 1 ;
The sum of the given infinite geometric series is 1. We used the formula S ∞ = 1 − r a with a = 2 1 and r = 2 1 . After substituting the values, we find that the total sum is 1.
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