The series you have given is a geometric series:

$S = \sum_{n=1}^{\infty} \left( \frac{1}{10} \right)^n$

This series is a geometric series, where the general form is:

$S = a + ar + ar^2 + ar^3 + \dots$

Where:

• $a$ is the first term.
• $r$ is the common ratio.

First Term:

From the given series, the first term ($a$) is:

$a = \left( \frac{1}{10} \right)^1 = \frac{1}{10}$

Common Ratio:

The common ratio ($r$) is the constant factor between successive terms, which is also:

$r = \frac{1}{10}$

Sum of the Infinite Series:

For a geometric series, the sum of an infinite series is given by the formula:

$S_{\infty} = \frac{a}{1 - r}$

This formula works when $|r| < 1$. In your case, $r = \frac{1}{10}$, which satisfies this condition, so we can use the formula:

$S_{\infty} = \frac{\frac{1}{10}}{1 - \frac{1}{10}} = \frac{\frac{1}{10}}{\frac{9}{10}} = \frac{1}{9}$

Thus, the sum of the infinite series is:

$S_{\infty} = \frac{1}{9}$

Partial Sums:

To calculate the partial sums of this series, we use the formula for the sum of the first $n$ terms of a geometric series:

$S_n = a \frac{1 - r^n}{1 - r}$

Substituting the values of $a = \frac{1}{10}$ and $r = \frac{1}{10}$:

$S_n = \frac{1}{10} \cdot \frac{1 - \left( \frac{1}{10} \right)^n}{1 - \frac{1}{10}} = \frac{1}{10} \cdot \frac{1 - \left( \frac{1}{10} \right)^n}{\frac{9}{10}} = \frac{1 - \left( \frac{1}{10} \right)^n}{9}$

Example of First Few Partial Sums:

• For $n = 1$, the partial sum is:

$S_1 = \frac{1 - \left( \frac{1}{10} \right)^1}{9} = \frac{1 - \frac{1}{10}}{9} = \frac{\frac{9}{10}}{9} = \frac{1}{10}$

• For $n = 2$, the partial sum is:

$S_2 = \frac{1 - \left( \frac{1}{10} \right)^2}{9} = \frac{1 - \frac{1}{100}}{9} = \frac{\frac{99}{100}}{9} = \frac{11}{100}$

• For $n = 3$, the partial sum is:

$S_3 = \frac{1 - \left( \frac{1}{10} \right)^3}{9} = \frac{1 - \frac{1}{1000}}{9} = \frac{\frac{999}{1000}}{9} = \frac{111}{1000}$

Summary:

• First term (a): $\frac{1}{10}$
• Common ratio (r): $\frac{1}{10}$
• Sum of the infinite series: $\frac{1}{9}$
• Partial sums:
  • $S_1 = \frac{1}{10}$
  • $S_2 = \frac{11}{100}$
  • $S_3 = \frac{111}{1000}$

Do you want to explore more partial sums or have any questions about the process?

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Further Questions:

1. What is the behavior of the partial sums as $n$ increases?
2. How do geometric series differ from arithmetic series?
3. What is the convergence criterion for a geometric series?
4. How does the sum of a geometric series change if $r > 1$?
5. How would you derive the sum of a geometric series with negative terms?

Tip: The convergence of a geometric series is highly dependent on the common ratio $r$. If $|r| < 1$, the series will converge; otherwise, it will diverge.