To evaluate the infinite geometric series described by the terms [tex]\(3 + 9 + 27 + 81 + \ldots\)[/tex], let's first identify the first term and the common ratio of the series.
1. Identify the first term:
The first term [tex]\(a\)[/tex] of the series is [tex]\(3\)[/tex].
2. Determine the common ratio:
The common ratio [tex]\(r\)[/tex] can be found by dividing any term by the previous term.
- For example, the second term [tex]\(9\)[/tex] divided by the first term [tex]\(3\)[/tex] gives [tex]\(r = \frac{9}{3} = 3\)[/tex].
- You can check this ratio with other terms as well: [tex]\[
\frac{27}{9} = 3 \text{ and } \frac{81}{27} = 3.
\][/tex]
Thus the common ratio [tex]\(r\)[/tex] is [tex]\(3\)[/tex].
3. Determine if the series converges or diverges:
For an infinite geometric series to have a sum, the absolute value of the common ratio [tex]\(r\)[/tex] must be less than [tex]\(1\)[/tex] ([tex]\(|r| < 1\)[/tex]). If [tex]\(|r| \geq 1\)[/tex], the series diverges and does not have a finite sum.
4. Check the common ratio:
Since [tex]\(r = 3\)[/tex], we have [tex]\(|3| = 3\)[/tex], which is greater than [tex]\(1\)[/tex].
Therefore, the infinite geometric series [tex]\(3 + 9 + 27 + 81 + \ldots\)[/tex] does not converge; it diverges.
The correct answer is:
No sum, divergent.
