QuestionFill in each blank. $$\begin{array}{l} 6+\frac{6}{5}+\frac{6}{25}+\frac{6}{125} \ldots \\ \lim _{n \rightarrow \infty} a_{n}= \\ \lim _{n \rightarrow \infty} S_{n}= \end{array}$$ $\square$ $\square$

Studdy Solution
Determine the sum of the infinite series $S_n$:
The sum of an infinite geometric series is given by:
$$S = \frac{a}{1-r} = \frac{6}{1-\frac{1}{5}} = \frac{6}{\frac{4}{5}} = 6 \times \frac{5}{4} = \frac{30}{4} = 7.5$$
Thus, $\lim_{n \rightarrow \infty} S_n = 7.5$.
The completed blanks are:
1. $\lim_{n \rightarrow \infty} a_n = 0$
2. $\lim_{n \rightarrow \infty} S_n = 7.5$