Which one of the following is the sum of the series 1 + 12 + 14 + to a) 23 b) 12 c) 1 d) 2

The given series is an infinite geometric series of the form: $$S=1+\frac{1}{2}+\frac{1}{4}+\dots$$

Step 1: Identify the first term and common ratio

• First term ($a$) = 1
• Common ratio ($r$) = $\frac{1}{2}$

Step 2: Check for convergence

Since $|r|=\frac{1}{2}<1$, the series converges.

Step 3: Apply the sum formula for an infinite geometric series

The sum of an infinite geometric series is given by: $$S_{\infty }=\frac{a}{1-r}$$

Step 4: Substitute the values

$$S_{\infty }=\frac{1}{1-\frac{1}{2}}=\frac{1}{\frac{1}{2}}=2$$

Thus, the sum of the series is 2. The correct option is (d) 2.