Geometric Series
sum a*r^n converges to a/(1-r) when |r|<1.
Example: sum (1/2)^n = 2.
Adaptive
Learn Infinite Sequences and Series
Read the notes, then try the practice. It adapts as you go.When you're ready.
Session Length
~17 min
Adaptive Checks
15 questions
Transfer Probes
6
Lesson Notes
This topic covers infinite sequences and series for AP Calculus BC Unit 10. Sequences converge or diverge; series are sums of infinite terms. Convergence tests (ratio, root, comparison, integral, alternating series) determine behavior.
Taylor and Maclaurin series approximate functions as power series. Error bounds quantify approximation accuracy. Key skills: convergence tests, Taylor/Maclaurin construction, radius/interval of convergence, Lagrange error bound.
You'll be able to:
- Explain the concept of geometric series and its role in infinite sequences and series
- Distinguish between p-series and ratio test in context
- Analyze how ratio test applies to real-world scenarios
- Apply how taylor series applies to real-world scenarios
- Evaluate how maclaurin series applies to real-world scenarios
One step at a time.
Interactive Exploration
Adjust the controls and watch the concepts respond in real time.
Key Concepts
p-Series
sum 1/n^p converges iff p>1.
Example: sum 1/n^2 converges; sum 1/n diverges.
Ratio Test
L = lim |a_{n+1}/a_n|. L<1: converges. L>1: diverges.
Example: sum n!/3^n: L=lim(n+1)/3=inf, diverges.
Taylor Series
f(x) = sum f^(n)(a)/n! (x-a)^n.
Example: e^x = sum x^n/n! about a=0.
Maclaurin Series
Taylor series centered at a=0.
Example: sin x = sum (-1)^n x^(2n+1)/(2n+1)!.
Radius of Convergence
R such that series converges for |x-a|<R.
Example: sum x^n/n!: R=infinity.
Lagrange Error Bound
|R_n(x)| <= M/(n+1)! |x-a|^(n+1).
Example: T_3 for e^x at x=1: error<=e/24.
Explore your way
Choose a different way to engage with this topic — no grading, just richer thinking.
Explore your way — choose one:
Concept Map
See how the key ideas connect. Nodes color in as you practice.
Worked Example
Walk through a solved problem step-by-step. Try predicting each step before revealing it.
Adaptive Practice
This is guided practice, not just a quiz. Hints and pacing adjust in real time.
Small steps add up.
What you get while practicing:
- Math Lens cues for what to look for and what to ignore.
- Progressive hints (direction, rule, then apply).
- Targeted feedback when a common misconception appears.
Ad Placeholder
Teach It Back
The best way to know if you understand something: explain it in your own words.
Keep Practicing
More ways to strengthen what you just learned.