AP Calculus BC
Advanced Placement Calculus BC including series, parametric equations, and polar functions.
Advanced Topics
Infinite Series and Convergence
Summing the Infinite
An infinite series adds up terms forever! But not all series add to a finite number. Understanding convergence means knowing when an infinite sum makes sense.
- A series converges if the sum approaches a real number as more terms are added.
- The geometric series \( \sum_{n=0}^\infty ar^n \) converges if \( |r| < 1 \).
- Tests like the Ratio Test, Root Test, and Integral Test help determine convergence.
Real-World Connections
Infinite series are key for representing complicated functions, like \( e^x \) or \( \sin x \), as power series.
Key Formula
\[S = \frac{a}{1 - r}\]
Examples
The sum \( 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots = 2 \).
The series \( \sum_{n=1}^\infty \frac{1}{n^2} \) converges to \( \frac{\pi^2}{6} \).
In a Nutshell
Infinite series can sum up forever—if they're convergent, the total is finite!