Sum the series to infinity :
`1 +(1)/(2) +(1)/(4) +(1)/(8) + ... `

Answer

To find the sum of the series to infinity: \(1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots\), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Series Type:**
The given series is \(1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots\). We need to determine if this is a geometric series.

2. **Check the Common Ratio:**
To confirm it's a geometric series, we need to check if the ratio of consecutive terms is constant.
\[ \frac{\frac{1}{2}}{1} = \frac{1}{2}, \quad \frac{\frac{1}{4}}{\frac{1}{2}} = \frac{1}{2}, \quad \frac{\frac{1}{8}}{\frac{1}{4}} = \frac{1}{2} \]
Since the ratio is constant and equal to \(\frac{1}{2}\), this is indeed a geometric series.

3. **Identify the First Term and Common Ratio:**
The first term \(a\) of the series is \(1\).
The common ratio \(r\) is \(\frac{1}{2}\).

4. **Sum of an Infinite Geometric Series:**
The formula for the sum \(S\) of an infinite geometric series \(a + ar + ar^2 + ar^3 + \ldots\) where \(|r| < 1\) is:
\[ S = \frac{a}{1 - r} \]

5. **Substitute the Values:**
Here, \(a = 1\) and \(r = \frac{1}{2}\).
\[ S = \frac{1}{1 - \frac{1}{2}} \]

6. **Simplify the Expression:**
\[ S = \frac{1}{\frac{1}{2}} = 2 \]

### Final Answer:
The sum of the series to infinity is \(2\).