To what number does the series $\sum \underset{}{k=0}\stackrel{}{\infty }{\left(-\frac{1}{2}\right)}^k$ converge?

• A. $0$
• B. $-2$
• C. $2$
• D. $1$

Answer: (C)

Verified step by step guidance

1. Recognize that the given series \( \sum_{k=0}^{\infty} \left(-\frac{1}{2}\right)^k \) is an infinite geometric series with the first term \( a = 1 \) (when \( k = 0 \)) and the common ratio \( r = -\frac{1}{2} \).
2. Recall the formula for the sum of an infinite geometric series: \( S = \frac{a}{1 - r} \), which is valid when \( |r| < 1 \).
3. Verify that \( |r| = \left| -\frac{1}{2} \right| = \frac{1}{2} < 1 \), so the formula for the sum of the series can be applied.
4. Substitute the values \( a = 1 \) and \( r = -\frac{1}{2} \) into the formula \( S = \frac{a}{1 - r} \). This gives \( S = \frac{1}{1 - (-\frac{1}{2})} \).
5. Simplify the denominator \( 1 - (-\frac{1}{2}) \) to \( 1 + \frac{1}{2} = \frac{3}{2} \), and then simplify the fraction \( \frac{1}{\frac{3}{2}} \) to find the sum of the series.