To determine if the sum of the series [tex]\(\frac{1}{27} + \frac{1}{3} + \frac{1}{3}\)[/tex] equals 243, let's proceed step-by-step through the relevant calculations.
1. Adding the Series Terms:
- First, we notice that the terms in the series are [tex]\(\frac{1}{27}\)[/tex], [tex]\(\frac{1}{3}\)[/tex], and [tex]\(\frac{1}{3}\)[/tex].
- To find the sum of the series, we add these three fractions together.
2. Common Denominator:
- We need a common denominator to add these fractions. The least common multiple of 27 and 3 is 27.
- Rewrite each fraction with a denominator of 27:
[tex]\[
\frac{1}{27} + \frac{1}{3} = \frac{1}{27} + \frac{9}{27}
\][/tex]
[tex]\[
\frac{1}{27} + \frac{9}{27} + \frac{9}{27} = \frac{1}{27} + \frac{9}{27} + \frac{9}{27} = \frac{1 + 9 + 9}{27} = \frac{19}{27}
\][/tex]
3. Summing the Fractions:
- Adding these fractions yields:
[tex]\[
\frac{19}{27}
\][/tex]
4. Decimal Representation:
- Converting [tex]\(\frac{19}{27}\)[/tex] to a decimal:
[tex]\[
\frac{19}{27} \approx 0.7037037037
\][/tex]
5. Comparison:
- We compare the sum [tex]\(0.7037037037\)[/tex] with 243.
Given the sum of the series is [tex]\(0.7037037037\)[/tex] and it does not match 243, we conclude that the sum of the given series [tex]\(\frac{1}{27} + \frac{1}{3} + \frac{1}{3}\)[/tex] is not equal to 243.
