What characterizes a convergent geometric series?

What characterizes a convergent geometric series?

• A. Its sum tends towards a fixed value
• B. Its terms oscillate indefinitely
• C. It consists of all integer values
• D. It diverges with increasing values

Answer: (A)

A convergent geometric series is characterized by its ability to approach a specific fixed sum as more terms are added. This characteristic occurs when the absolute value of the common ratio (the factor by which consecutive terms are multiplied) is less than one. In such cases, the sum of an infinite number of terms can be calculated using the formula \( S = \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio. As you keep adding terms, the total sum converges toward this fixed value rather than fluctuating or growing indefinitely. In contrast, a series that oscillates indefinitely does not settle on a fixed sum and can vary significantly. A geometric series consisting of all integer values would not necessarily imply convergence, as the common ratio could lead to divergence. Similarly, a series that diverges would not be considered convergent, as it grows larger without approaching any particular value. Therefore, the defining characteristic of a convergent geometric series is that it tends towards a fixed value, solidifying the correctness of the answer.

A convergent geometric series is characterized by its ability to approach a specific fixed sum as more terms are added. This characteristic occurs when the absolute value of the common ratio (the factor by which consecutive terms are multiplied) is less than one. In such cases, the sum of an infinite number of terms can be calculated using the formula ( S = \frac{a}{1 - r} ), where ( a ) is the first term and ( r ) is the common ratio. As you keep adding terms, the total sum converges toward this fixed value rather than fluctuating or growing indefinitely.

In contrast, a series that oscillates indefinitely does not settle on a fixed sum and can vary significantly. A geometric series consisting of all integer values would not necessarily imply convergence, as the common ratio could lead to divergence. Similarly, a series that diverges would not be considered convergent, as it grows larger without approaching any particular value. Therefore, the defining characteristic of a convergent geometric series is that it tends towards a fixed value, solidifying the correctness of the answer.