Power Series Convergence Finder
Find the Radius (R) and Interval of convergence for Σ cn(x – a)n
Define Series
Use ‘n’ as the variable. Ex: 1/(n*2^n)
The series is centered at x = a.
Ratio Limit L
Radius of Convergence R
Interval of Convergence
Convergence Visualizer
Visual: The Green Zone represents x-values where the series gives a finite result. Outside this zone, the sum explodes to infinity.
Understanding Power Series Convergence
A Power Series is essentially an infinite polynomial. While polynomials (like $x^2 + 2$) work for any x, infinite series might only work (converge) for a specific range of x values. This range is called the Interval of Convergence.
The Ratio Test
The most common way to find this interval is the Ratio Test. We look at the limit of the ratio of consecutive coefficients:
L = limn→∞ | cn+1 / cn |
Radius of Convergence (R)
The “size” of the safe zone is determined by R.
- R = 1/L (if L is finite and non-zero)
- R = ∞ (if L = 0, converges everywhere)
- R = 0 (if L = ∞, converges only at center)
Geometric Interpretation
The interval is always centered at a. It stretches out distance R to the left and right.
Interval = (a – R, a + R)
Note: The Ratio Test is inconclusive at the exact endpoints of the interval. You must test the specific x-values (a-R) and (a+R) individually using other tests (Alternating Series, p-Series) to see if brackets [ ] or parentheses ( ) are needed.
Frequently Asked Questions (FAQ)
Q: What does “Convergence” mean?
A: It means the infinite sum adds up to a specific, finite number. If it “Diverges”, the sum grows to infinity or oscillates forever without settling.
Q: Why check endpoints manually?
A: At the very edge of the radius (distance R), the terms might shrink just fast enough to add up, or they might stay just big enough to explode. The Ratio Test simply returns “1” (Inconclusive) at these points.
Q: What if R = Infinity?
A: This is the best case! It means the series works for ALL real numbers (Interval: -∞ to ∞). Common examples are sin(x), cos(x), and e^x.