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  2. AP Calculus
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What is a geometric series?

A series with a constant ratio between successive terms.

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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.

General form of a geometric series (starting from n=0)?

∑n=0∞acdotrn\sum_{n=0}^{\infty} a cdot r^nn=0∑∞​acdotrn

General form of a geometric series (starting from n=1)?

∑n=1∞acdotrn−1\sum_{n=1}^{\infty} a cdot r^{n-1}n=1∑∞​acdotrn−1

Formula for the sum of a converging geometric series?

∑n=1∞acdotrn−1=∑n=0∞acdotrn=a1−r\sum_{n=1}^{\infty} acdot r^{n-1} = \sum_{n=0}^{\infty} a cdot r^n = \frac{a}{1-r}n=1∑∞​acdotrn−1=n=0∑∞​acdotrn=1−ra​

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.