AP Calculus BC Flashcards: Working with Geometric Series

Question 1

For which AP Calculus exam is the convergence of geometric series a required topic?

Accepted Answer

The convergence and sum of a geometric series is a topic exclusive to the AP Calculus BC exam.

Question 2

What is a geometric series?

Accepted Answer

A geometric series is a series that has a constant ratio between successive terms. (BC ONLY)

Question 3

Does the series $\sum_{n=0}^{\infty} 4 (\frac{1}{2})^n$ converge or diverge?

Accepted Answer

The series converges because it is a geometric series with a constant ratio $r = \frac{1}{2}$, and $|r| < 1$.

Question 4

What is the formula for the sum of a convergent geometric series?

Accepted Answer

For a series $\sum_{n=0}^{\infty} a r^n$ where $|r|<1$, the sum is $S = \frac{a}{1-r}$, where 'a' is the first term. (BC ONLY)

Question 5

Does the series $\sum_{n=0}^{\infty} (\frac{4}{3})^n$ converge or diverge?

Accepted Answer

The series diverges because it is a geometric series with a constant ratio $r = \frac{4}{3}$, and $|r| \geq 1$.

Question 6

Determine if the series $\sum_{n=0}^{\infty} 2(-\frac{3}{5})^n$ converges, and if so, find its sum.

Accepted Answer

The series converges because $|r| = |-\frac{3}{5}| < 1$. The sum is $\frac{a}{1-r} = \frac{2}{1 - (-3/5)} = \frac{2}{8/5} = \frac{10}{8} = 1.25$.

Question 7

Find the sum of the series $\sum_{n=0}^{\infty} 3 (\frac{1}{4})^n$.

Accepted Answer

The series converges to $\frac{a}{1-r} = \frac{3}{1 - 1/4} = \frac{3}{3/4} = 4$. The first term 'a' is 3 and the ratio 'r' is 1/4.

Question 8

Under what condition does a geometric series converge?

Accepted Answer

A geometric series converges if the absolute value of its constant ratio, |r|, is less than 1 (i.e., |r| < 1).

Question 9

In the geometric series form $\sum_{n=0}^{\infty} a r^n$, what do 'a' and 'r' represent?

Accepted Answer

'a' represents the first term of the series, and 'r' represents the constant ratio between terms.

Question 10

What is the primary test used to determine if a geometric series converges or diverges?

Accepted Answer

The primary test is to check the value of the constant ratio 'r'; the series converges if $|r| < 1$ and diverges if $|r| \geq 1$.

All Flashcards (10)