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A geometric series is a sequence of terms where each term after the first is found by multiplying the previous term by a fixed, non-zero number known as the common ratio. This type of series can be written in a general form

• \( a + ar + ar^2 + ar^3 + \dots \,\)

where \( a \) is the first term and \( r \) is the common ratio. When we talk about an infinite geometric series, it means that there are an infinite number of terms. The convergence or divergence of a geometric series mainly depends on the value of \( r \).
If \( |r| < 1 \), the series converges, and if \( |r| \geq 1 \), the series diverges.
In our example, the series \( \sum_{n=0}^{\infty} \left(\frac{1}{\sqrt{2}}\right)^{n} \) is geometric with a common ratio of \( \frac{1}{\sqrt{2}} \). Since \( \left| \frac{1}{\sqrt{2}} \right| < 1 \), the series converges.

An infinite series involves the sum of infinitely many terms. It's defined not merely as a sum but as a limit of a sequence of partial sums. Understanding infinite series helps in exploring important concepts in calculus and analysis.
An infinite series is often written as \( \sum_{n=0}^{\infty} a_n \), where \( a_n \) represents the sequence terms. To determine whether an infinite series converges, we observe whether the sequence of partial sums has a finite limit. For geometric series, particularly, the rule \( |r| < 1 \) indicates convergence.
Our exercise deals with such an infinite series where terms are determined by a geometric progression. Given its common ratio of \( \frac{1}{\sqrt{2}} \), the series converges because the ratio is effectively captured within the range that ensures convergence.

The sum of a convergent series can be an intriguing topic, especially when evaluating infinite series. For a geometric series that meets the convergence condition \( |r| < 1 \), the sum is calculated using the formula

• \( \frac{a}{1-r} \)

In this formula, \( a \) is the initial term, and \( r \) is the common ratio. In our exercise, \( a = 1 \) for \( a^0 = 1 \), and \( r = \frac{1}{\sqrt{2}} \).
This formula promptly gives us the sum \( \frac{1}{1 - \frac{1}{\sqrt{2}}} \). After simplifying, the calculation steps yield a sum of \( 2 + \sqrt{2} \).
This result confirms that despite dealing with infinitely many terms, when adhering to conditions of convergence, even infinite series can be nicely boiled down to a finite sum!