Question

What is the sum of an infinite geometric series with first term equal to \(1\) and common ratio \(\frac{1}{2}\)?

Hint:

For an infinite geometric series, the sum exists only if \(|r| < 1\). In that case, use \(S_{\infty} = \frac{a}{1-r}\).

Answer 1

Solution:
Step 1 (Formula for the sum of infinite GP).
If \(|r| < 1\), the sum to infinity is: \[ S_{\infty} = \frac{a}{1 - r} \] where \(a\) = first term, \(r\) = common ratio. Step 2 (Substitute given values).
Here \(a = 1\), \(r = \frac{1}{2}\): \[ S_{\infty} = \frac{1}{1 - \frac{1}{2}} = \frac{1}{\frac{1}{2}} = 2. \] Step 3 (Interpretation).
The sum converges because \(|r| = \frac{1}{2} < 1\). This means adding all terms forever (1, 0.5, 0.25, 0.125, ...) approaches exactly \(2\). \[ {2} \]