V - Power series

1.1 Radius of convergence

let Σ^∞_n=0 a_n(x-x₀)ⁿ be a power series

there exists a unique 0 ≤ R ≤ ∞ such that the power series converges absolutely for |x - x₀| < R and diverges for |x - x₀| > R

R is the radius of convergence of the power series

1.2 Convergence

let y =/= x₀ such that Σ^∞_n=0 a_n(y-x₀)ⁿ converges

then Σ^∞_n=0 a_n(x-x₀)ⁿ is absolutely convergent for all x ∈ IR with |x-x₀| < |y-x₀|

1.3 Root test

suppose a_n =/= 0 for n sufficiently large and lim |a_n|^1/n = 1/R

then Σ^∞_n=0 a_n(x-x₀)ⁿ has radius of convergence R

1.4 Ratio test

suppose a_n =/= 0 for n sufficiently large and lim (|a_n+1|/|a_n|) = 1/R

then Σ^∞_n=0 a_n(x-x₀)ⁿ has radius of convergence R