write this series in sigma notation 1/3 + 1/9  1/27 + 1/81+ ... them after evaluate the first four partial sums, S1,S2,S3,S4, and the limit of the partial sums.

The given series is:

\[
\frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{81} + \dots
\]

This is a geometric series where the first term \(a = \frac{1}{3}\) and the common ratio \(r = \frac{1}{3}\).

### Step 1: Sigma Notation

The series can be written in sigma notation as:

\[
\sum_{n=1}^{\infty} \frac{1}{3^n}
\]

### Step 2: Evaluating the First Four Partial Sums

The partial sums \(S_n\) represent the sum of the first \(n\) terms of the series.

- \(S_1 = \frac{1}{3}\)
- \(S_2 = \frac{1}{3} + \frac{1}{9} = \frac{3 + 1}{9} = \frac{4}{9}\)
- \(S_3 = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} = \frac{9 + 3 + 1}{27} = \frac{13}{27}\)
- \(S_4 = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{81} = \frac{27 + 9 + 3 + 1}{81} = \frac{40}{81}\)

### Step 3: Limit of the Partial Sums

For an infinite geometric series, the sum can be found using the formula:

\[
S_{\infty} = \frac{a}{1 - r} = \frac{\frac{1}{3}}{1 - \frac{1}{3}} = \frac{\frac{1}{3}}{\frac{2}{3}} = \frac{1}{2}
\]

Thus, the limit of the partial sums as \(n \to \infty\) is:

\[
S_{\infty} = \frac{1}{2}
\]

### Summary:

- \(S_1 = \frac{1}{3}\)
- \(S_2 = \frac{4}{9}\)
- \(S_3 = \frac{13}{27}\)
- \(S_4 = \frac{40}{81}\)
- The limit \(S_{\infty} = \frac{1}{2}\)

Would you like further details on how the sums are calculated or any other questions?

Here are some relative questions for further exploration:
1. How do you derive the sum formula for an infinite geometric series?
2. What happens if the common ratio \(r\) is greater than 1?
3. How can you use sigma notation for a series with a different starting index?
4. What is the importance of the common ratio in determining convergence?
5. How do geometric series apply in real-world scenarios like finance or physics?

**Tip:** A geometric series converges only if the absolute value of the common ratio \(r\) is less than 1.

write this series in sigma notation 1/3 + 1/9  1/27 + 1/81+ ... then evaluate the first four partial sums, S1, S2, S3, S4, and the limit of the partial sums.

Understand how to express the series 1/3 + 1/9 + 1/27 + ... in sigma notation. Evaluate the first four partial sums and find the limit of the series using geometric series formulas. This guide explores key concepts in geometric series and is ideal for students in Grades 10-12.

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