91Ó°ÊÓ

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For the series you're working with, the first term, denoted as **a**, is 1, and the common ratio, **r**, is \(\frac{9}{10}\). This pattern continues up to \(\left(\frac{9}{10}\right)^{n}\).

The formula for the sum of a geometric series is crucial. It's given by:

\[ S = a \cdot \frac{1 - r^{n+1}}{1 - r} \]

For this specific series, substituting the values, the formula simplifies to:

\[ S = \frac{1 - \left(\frac{9}{10}\right)^{n+1}}{\frac{1}{10}} \]

This formula efficiently calculates the total sum by accounting for all terms in the series.

A closed form expression provides a way to write a complex mathematical expression simply and compactly. For a geometric series, this means expressing the entire sum without listing out each term individually. We achieve this with the formula:

\[ S = 10 - 10 \cdot \left(\frac{9}{10}\right)^{n+1} \]

This representation encompasses all elements of the series and allows for easy computation. Understanding closed forms helps in quickly evaluating sums without manual repetition of terms.

It's a more elegant and less labor-intensive approach that leverages mathematical properties to simplify processes.

Sometimes, we need an actual numerical value rather than just a formula. This is where numerical approximation comes into play. With the closed form already at hand, you simply plug in the value of **n** into:

\[ S = 10 - 10 \cdot \left(\frac{9}{10}\right)^{n+1} \]

Then, the computation gives you a specific numerical result based on your chosen **n**. After calculation, you can round this value to three decimal places for precision.

For instance, by setting **n=5** in the equation, you'll find an approximate sum of 9.411. Experimenting with different values of **n** will show how the sum converges closer to 10 as **n** increases.