Question: What is the value of \( S_4 \) for this series? \[ \sum_{n=1}^{\infty} 6(2)^{n-1} \] 96 48…

What is the value of \( S_4 \) for this series?

\[ \sum_{n=1}^{\infty} 6(2)^{n-1} \]

96

48

90

84

Solution

The series given is \[ \sum_{n=1}^{\infty} 6(2)^{n-1} \] This is an infinite geometric series where the first term \( a \) is 6 and the common ratio \( r \) is 2. The sum \( S \) of an infinite geometric series is calculated using the formula: \[ S = \frac{a}{1 - r} \] In this case, \( r = 2 \), and since the absolute value of the common ratio must be less than 1 for the sum to converge, this series does not converge. Thus, the sum \( S \) is undefined, as the series diverges. If there is any additional context needed to interpret \( S_4 \) specifically, please provide more details.