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Power Series: Interval of Convergence
AP Calculus BC · Unit 10 Infinite Series
Learning Objectives
Find radius of convergence R via Ratio Test
Determine the open interval |x−c| < R
Test endpoints separately (p-series, AST)
Write the final interval (brackets vs parens)
Ratio Test
L = lim |a_{n+1}/a_n|
L < 1 → Absolutely Converges
L > 1 → Diverges
L = 1 → Test Inconclusive
Solve |a_{n+1}/a_n| < 1 for x to find the open interval, then check endpoints.
Tags
power series
ratio test
convergence
interval
POWER SERIES ∑ aₙ(x−c)ⁿ
Σ xⁿ/n! (e^x, R=∞)
Σ xⁿ (Geometric, R=1)
Σ xⁿ/n (ln series, R=1)
Σ (−1)ⁿ x²ⁿ/(2n)! (cos x, R=∞)
Σ nⁿxⁿ/n! (R=1/e)
Number of Terms N
10
Evaluation Point x
0.00
Radius of Convergence R
—
Interval of Convergence
—
Partial Sum S_N(x)
—
Exact Value f(x)
—
Inside the interval, the partial sum S_N(x) converges to f(x) as N→∞. Outside, it diverges wildly.