Glossary
Common Ratio (r)
The constant factor by which each term in a geometric series is multiplied to get the next term, denoted by 'r'.
Example:
For the series , the Common Ratio (r) is .
Convergent Series
An infinite series whose sequence of partial sums approaches a finite limit. For a geometric series, this occurs when $0 < |r| < 1$.
Example:
The series is a Convergent Series because its sum approaches a finite value of 2.
Divergent Series
An infinite series whose sequence of partial sums does not approach a finite limit, meaning it either goes to infinity, negative infinity, or oscillates. For a geometric series, this occurs when $|r| \geq 1$.
Example:
The series is a Divergent Series because its sum grows infinitely large.
Geometric Series
An infinite series where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. It typically takes the form $\sum_{n=0}^{\infty} a \cdot r^n$ or $\sum_{n=1}^{\infty} a \cdot r^{n-1}$.
Example:
The series is a Geometric Series because each term is twice the previous one, with and .
Geometric Series Test
A theorem used to determine if a geometric series converges or diverges based on the absolute value of its common ratio, 'r'. It states that a geometric series converges if 0 < |r| < 1 and diverges if |r| ≥ 1.
Example:
To check if the series converges, we apply the Geometric Series Test by observing that , confirming its convergence.
Initial Term (a)
The first term in a geometric series, denoted by 'a', which sets the starting value of the sequence.
Example:
In the series , the Initial Term (a) is 7.
Sum of the Series
The finite value that a convergent infinite series approaches. For a convergent geometric series, it is calculated using the formula $\frac{a}{1-r}$.
Example:
For the geometric series , the Sum of the Series is .