Question

What is the sum of the series $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots$? 
 

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

An infinite geometric series converges only if $|r|<1$, and then $S = \frac{a}{1-r}$.

Answer 1

The Correct Option is B

The given series is an infinite geometric series: \( S = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots \).

This series is geometric because each term is a fixed fraction, called the common ratio, of the previous term.

For a geometric series, the sum \( S \) of an infinite number of terms where \( |r| < 1 \) is given by the formula:

\( S = \frac{a}{1-r} \)

 

where \( a \) is the first term of the series and \( r \) is the common ratio.

In this series, \( a = 1 \) and \( r = \frac{1}{2} \).

Plugging these values into the formula gives us:

\( S = \frac{1}{1-\frac{1}{2}} = \frac{1}{\frac{1}{2}} = 2 \)

 

Therefore, the sum of the series is 2