Key Takeaways
Lesson Description
Key Takeaways
• Infinite sequences approach a limiting value or diverge.
• Infinite series represent the sum of infinitely many terms.
• Convergence Tests determine whether a series converges.
• Taylor and Maclaurin series approximate functions using polynomials.
• Radius and interval of convergence determine where a power series is valid.
• Taylor polynomials become more accurate as more terms are added.
Must Know Equations
Geometric Series:
Σ arⁿ = a/(1-r), |r|<1
nth Term Test:
If lim an ≠ 0, the series diverges.
Maclaurin Series for eˣ:
1 + x + x²/2! + x³/3! + ...
Maclaurin Series for sin(x):
x − x³/3! + x⁵/5! − ...
Maclaurin Series for cos(x):
1 − x²/2! + x⁴/4! − ...
Must Know Vocabulary
Sequence
Series
Convergence
Divergence
Geometric Series
Power Series
Taylor Series
Maclaurin Series
Radius of Convergence
Interval of Convergence
AP Exam Focus
• Determine whether a series converges or diverges.
• Apply Ratio, Integral, Comparison, Alternating Series, and p-Series Tests.
• Construct Taylor and Maclaurin polynomials.
• Find intervals and radii of convergence.