All Flashcards
What is a geometric series?
A series with a constant ratio between successive terms.
What is 'a' in a geometric series?
'a' is the initial term of the geometric series.
What is 'r' in a geometric series?
'r' is the common ratio between consecutive terms in a geometric series.
What does it mean for a series to converge?
A series converges if its partial sums approach a finite limit.
What does it mean for a series to diverge?
A series diverges if its partial sums do not approach a finite limit.
What does the Geometric Series Test state?
A geometric series converges if 0 < |r| < 1 and diverges if |r| ≥ 1, where 'r' is the common ratio.
Explain the significance of the common ratio 'r' in determining convergence.
The absolute value of 'r' determines convergence/divergence: if 0 < |r| < 1, the series converges; if |r| ≥ 1, the series diverges.
How does changing the starting index of a geometric series affect its convergence?
Changing the starting index (e.g., from n=0 to n=1) does not affect whether the series converges or diverges.
What is the key idea behind the geometric series test?
The geometric series test provides a straightforward condition based on the common ratio 'r' to determine if an infinite geometric series converges or diverges.