To determine whether the given sequence [tex]\( b_n = 10 \left(\frac{1}{3}\right)^{n-1} \)[/tex] converges and, if so, to what value, we will analyze it as a geometric series. Here are the steps:
1. Identify the general form of the series:
The given sequence [tex]\( b_n \)[/tex] can be rewritten as:
[tex]\[
b_n = 10 \left(\frac{1}{3}\right)^{n-1}
\][/tex]
This represents a geometric series of the form [tex]\( a \cdot r^{n-1} \)[/tex] where:
- [tex]\( a = 10 \)[/tex] (the first term)
- [tex]\( r = \frac{1}{3} \)[/tex] (the common ratio)
2. Criterion for convergence:
A geometric series converges if and only if the absolute value of the common ratio [tex]\( |r| \)[/tex] is less than 1.
- In this case, [tex]\( r = \frac{1}{3} \)[/tex]
- Since [tex]\( \left|\frac{1}{3}\right| < 1 \)[/tex], the series converges.
3. Formula for the sum of an infinite geometric series:
The sum [tex]\( S \)[/tex] of an infinite geometric series with the first term [tex]\( a \)[/tex] and common ratio [tex]\( r \)[/tex] (where [tex]\( |r| < 1 \)[/tex]) is given by:
[tex]\[
S = \frac{a}{1 - r}
\][/tex]
4. Apply the sum formula:
We substitute [tex]\( a = 10 \)[/tex] and [tex]\( r = \frac{1}{3} \)[/tex] into the formula:
[tex]\[
S = \frac{10}{1 - \frac{1}{3}} = \frac{10}{\frac{2}{3}} = 10 \cdot \frac{3}{2} = 15
\][/tex]
5. Conclusion:
Hence, the series [tex]\( \sum_{n=1}^\infty 10 \left(\frac{1}{3}\right)^{n-1} \)[/tex] converges to 15.
Therefore, the correct statement is:
- The series (sum) converges to 15.
