To determine whether the given series [tex]\(1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \ldots\)[/tex] has a limit and to find the limit if it exists, we need to analyze the series.
1. Identify the Series:
The series [tex]\(1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \ldots\)[/tex] is a geometric series. A geometric series is one where each term after the first is found by multiplying the previous term by a constant called the common ratio.
2. Determine the First Term and Common Ratio:
- The first term ([tex]\(a\)[/tex]) of this series is 1.
- The common ratio ([tex]\(r\)[/tex]) is the ratio of any term to the previous term. Here, [tex]\(r = \frac{1}{3}\)[/tex].
3. Check for Convergence:
A geometric series converges if the absolute value of the common ratio is less than 1 ([tex]\(|r| < 1\)[/tex]). In this case, [tex]\(|r| = \left| \frac{1}{3} \right| = \frac{1}{3} < 1\)[/tex].
4. Calculate the Sum of the Infinite Geometric Series:
For a convergent geometric series, the sum [tex]\(S\)[/tex] of the infinite series can be found using the formula:
[tex]\[
S = \frac{a}{1 - r}
\][/tex]
where [tex]\(a\)[/tex] is the first term and [tex]\(r\)[/tex] is the common ratio.
5. Substitute the Values:
- First term [tex]\(a = 1\)[/tex]
- Common ratio [tex]\(r = \frac{1}{3}\)[/tex]
Substituting these values into the formula gives:
[tex]\[
S = \frac{1}{1 - \frac{1}{3}}
\][/tex]
6. Simplify the Expression:
[tex]\[
S = \frac{1}{\frac{3}{3} - \frac{1}{3}} = \frac{1}{\frac{2}{3}} = 1 \cdot \frac{3}{2} = \frac{3}{2}
\][/tex]
7. Conclusion:
The series [tex]\(1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \ldots\)[/tex] converges and its sum is:
[tex]\[
\frac{3}{2} \approx 1.5
\][/tex]
Which, when calculated, results approximately in:
[tex]\[
1.4999999999999998
\][/tex]
Therefore, the given series has a limit and the limit is approximately [tex]\(1.4999999999999998\)[/tex].
