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.