Geometric Series

Definition

A geometric series is a type of infinite series where each term is a constant multiple of the previous term. The series follows a pattern of multiplying each term by a common ratio to get the next term in the sequence.

5 Must Know Facts For Your Next Test

1. The general formula for the $n^{\text{th}}$ term of a geometric series is $a_n = a_1 \cdot r^{n-1}$, where $a_1$ is the first term and $r$ is the common ratio.
2. The formula for the sum of the first $n$ terms of a geometric series is $S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$.
3. A geometric series converges (has a finite sum) if the common ratio $|r| < 1$, and diverges (has an infinite sum) if $|r| \geq 1$.
4. The formula for the sum of an infinite geometric series is $S = \frac{a_1}{1 - r}$ if $|r| < 1$.
5. Geometric series have many applications in mathematics, including in finance, physics, and computer science.

Review Questions

• Explain the relationship between the common ratio and the convergence or divergence of a geometric series.
  • The common ratio, $r$, is a critical factor in determining whether a geometric series converges (has a finite sum) or diverges (has an infinite sum). If the absolute value of the common ratio, $|r|$, is less than 1, then the series will converge. This is because as the number of terms increases, the terms in the series will become smaller and smaller, and the sum of the series will approach a finite value. Conversely, if $|r| \geq 1$, then the series will diverge, as the terms will either remain constant or grow larger without bound, resulting in an infinite sum.
• Describe the formula for the sum of the first $n$ terms of a geometric series and explain how it is derived.
  • The formula for the sum of the first $n$ terms of a geometric series is $S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$, where $a_1$ is the first term and $r$ is the common ratio. This formula is derived by recognizing that each term in the series is a multiple of the previous term by the common ratio $r$. By summing the geometric series, we can express the sum in terms of the first term and the common ratio, resulting in the given formula. This formula is useful for calculating the partial sums of a geometric series, which can be important in various applications.
• Explain how the formula for the sum of an infinite geometric series is derived and discuss the conditions under which this formula is valid.
  • The formula for the sum of an infinite geometric series is $S = \frac{a_1}{1 - r}$, where $a_1$ is the first term and $r$ is the common ratio. This formula is derived by recognizing that the sum of an infinite geometric series can be expressed as the limit of the partial sums as the number of terms approaches infinity. By manipulating the formula for the partial sum, we can show that the sum of an infinite geometric series converges to the given formula, but only if the absolute value of the common ratio, $|r|$, is less than 1. This condition ensures that the terms in the series become smaller and smaller, allowing the series to have a finite sum. If $|r| \geq 1$, the series will diverge, and the formula for the sum of an infinite geometric series is not valid.

Related terms

Common Ratio: The constant ratio between consecutive terms in a geometric series.

Infinite Series: A series that continues indefinitely, with no final term.

Partial Sum: The sum of the first n terms in a series.