← Unit 10: Infinite Sequences and Series (BC)
Infinite Series: Convergence Tests & Taylor Series (BC)
A series converges if its sequence of partial sums approaches a finite limit. Several tests (geometric, p-series, ratio, alternating series) determine convergence; Taylor series approximate functions using polynomials built from derivatives at a point.
Key convergence tests
Geometric series Σarⁿ converges iff |r|<1, to a/(1-r). p-series Σ1/nᵖ converges iff p>1. The Ratio Test (compute lim |a_(n+1)/a_n|) is powerful for series with factorials or exponentials — converges if the limit is <1, diverges if >1, inconclusive if =1.
Taylor series represent a function as an infinite polynomial built from its derivatives at a center point a: f(x) = Σ [f⁽ⁿ⁾(a)/n!]·(x-a)ⁿ. A Maclaurin series is just a Taylor series centered at a=0.
Key terms (2)
- Ratio Test
- For Σaₙ, compute L=lim|a_(n+1)/aₙ|; converges (absolutely) if L<1, diverges if L>1.
- Taylor series
- f(x) = Σ [f⁽ⁿ⁾(a)/n!](x-a)ⁿ, an infinite polynomial approximation centered at x=a.
Practice Quiz
Question 1 of 1Score so far: 0/0
Does Σ (1/2)ⁿ (from n=0 to ∞) converge?