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A geometric series is a series of the form \( a + ar + ar^2 + ar^3 + \ldots \), where \( a \) is the first term and \( r \) is the common ratio. This series is a fundamental concept in mathematics.
It converges if the absolute value of the common ratio \( r \) is less than 1. In a convergent geometric series, the sum can be calculated using \( \frac{a}{1-r} \).

• If \(|r| < 1\), the series converges.
• If \(|r| \geq 1\), the series diverges.

In our exercise, \( \sum_{n=0}^{\infty} \left( \frac{2}{10} \right)^n \) represents a geometric series with \( a = 2 \) and \( r = \frac{1}{10} \). Because \( \left| \frac{1}{10} \right| < 1 \), it converges.

Absolute convergence refers to a series \( \sum a_n \) that converges even when all its terms are replaced by their absolute values, i.e., \( \sum |a_n| \). When a series converges absolutely, it ensures the series will still converge regardless of the arrangement of its terms.
For example, if \( \sum_{n=0}^{\infty} a_n \) converges absolutely, \( \sum_{n=0}^{\infty} |a_n| \) will also converge. This property often simplifies analysis since absolute convergence implies regular convergence, but not vice versa.
In our situation, the series \( \sum_{n=0}^{\infty} \frac{1+\sin n}{10^{n}} \) was shown to absolutely converge as the comparable geometric series \( \sum_{n=0}^{\infty} \frac{2}{10^n} \) converges.

An alternating series is a series whose terms alternate in sign. An example is \( \sum (-1)^n a_n \), where terms oscillate between positive and negative.
For convergence, the alternating series test states the series converges if:

• The absolute value of the terms \( a_n \) decreases monotonically (getting smaller with each term).
• The limit of \( a_n \) as \( n \to \infty \) is zero.

Even though our series had an expression including the sine function, its main property checked was absolute convergence, not just alternating series rules, given the presence of \( \sin n \) in every term.

The sine function, denoted as \( \sin \), is a periodic function known for its oscillating behavior between -1 and 1. It is fundamental in trigonometry and wave analysis.
Within trigonometry, \( \sin \theta \) relates to the ratio of the opposite side to the hypotenuse in a right triangle.
In infinite series, sine functions like \( \sin n \) can introduce alternating characteristics as seen in our series \( \sum_{n=0}^{\infty} \frac{1+\sin n}{10^{n}} \), where the sine component adds variability to each term's sign.

An infinite series is the summation of an infinite sequence of numbers written as \( a_1 + a_2 + a_3 + \ldots \). The main concern with infinite series is whether they lead to a finite result, or "converge."
A series converges if the partial sums, when added continuously, approach a specific limit. Conversely, in divergence, partial sums grow unbounded.

• Convergence can depend on various tests - absolute convergence, ratio tests, etc.
• Handling infinity requires careful consideration of terms and limits.

For instance, our series \( \sum_{n=0}^{\infty} \frac{1+\sin n}{10^{n}} \) presented convergence due to both the behavior of its terms and comparison to a geometric series.