1. Home
  2. >AP Calculus BC
  3. >Sequence and Series
Help

Convergence & divergence of geometric series

Everything You Need in One Place

Homework problems? Exam preparation? Trying to grasp a concept or just brushing up the basics? Our extensive help & practice library have got you covered.

Learn and Practice With Ease

Our proven video lessons ease you through problems quickly, and you get tonnes of friendly practice on questions that trip students up on tests and finals.

Instant and Unlimited Help

Our personalized learning platform enables you to instantly find the exact walkthrough to your specific type of question. Activate unlimited help now!

0/1
?
Intros
Lessons
  1. Geometric Series Overview:
0/7
?
Examples
Lessons
  1. Convergence of Geometric Series
    Show that the following series are convergent and find its sum:
    1. ∑n=0∞13n\sum_{n=0}^{\infty} \frac{1}{3^n}
    2. ∑n=1∞[(−58)n−1+(1+3n7n)] \sum_{n=1}^{\infty} [(-\frac{5}{8})^{n-1}+(\frac{1+3^n}{7^n})]
    3. ∑n=0∞4n+223−4n\sum_{n=0}^{\infty}4^{n+2}2^{3-4n}
    4. ∑n=0∞42(n+2)53n−1\sum_{n=0}^{\infty} \frac{4^{2(n+2)}}{5^{3n-1}}
  2. Divergence of Geometric Series
    Show that the following series are divergent:
    1. ∑n=0∞3n−12n \sum_{n=0}^{\infty} \frac{3^{n-1}}{2^n}
    2. ∑n=0∞3n+223−n\sum_{n=0}^{\infty}3^{n+2}2^{3-n}
    3. ∑n=0∞[(14)n+(32)n2n]\sum_{n=0}^{\infty}[(\frac{1}{4})^n+(\frac{3}{2})^n2^n]
Topic Notes
?
In this section, we will take a look at the convergence and divergence of geometric series. We've learned about geometric sequences in high school, but in this lesson we will formally introduce it as a series and determine if the series is divergent or convergent. For the first few questions we will determine the convergence of the series, and then find the sum. For the last few questions, we will determine the divergence of the geometric series, and show that the sum of the series is infinity.
Formulas for Geometric Series:

∑n=0∞arn=a1−r\sum_{n=0}^{\infty}ar^n=\frac{a}{1-r} if -1 < rr < 1
∑n=1∞arn−1=a1−r\sum_{n=1}^{\infty}ar^{n-1}=\frac{a}{1-r} if -1 < rr < 1
If -1 < rr < 1, then the geometric series converges. Otherwise, the series diverges.
Basic Concepts
?
Related Concepts
?