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1. For each of the following power series, find the radius of convergence R and the interval of convergence (IOC) (prove your answers). (a) (b) 8 n=1 3η η -xn ∞ Σ()" n=1

1. For each of the following power series, find the radius of convergence R and the interval of convergence (IOC) (prove your answers). (a) (b) 8 n=1 3η η -xn ∞ Σ()" n=1

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Real Analysis II: Kindly find steps/examples for guide in photo II
1. For each of the following power series, find the radius of convergence R and the interval of convergence (IOC) (prove your answers). (a) (b) 8 n=1 3η η -xn ∞ Σ()" n=1
Transcribed Image Text:1. For each of the following power series, find the radius of convergence R and the interval of convergence (IOC) (prove your answers). (a) (b) 8 n=1 3η η -xn ∞ Σ()" n=1
EX. (Boundary points = So (im |an| = lim |+| ns anti EX. Find R.O.C. Notice an = and R=1. check Xx=1: = check x = -1 = f(x) = f(-1) = 28 (-1) ² = 1 - 1 + |-|-|-=-=- C = (-1, 1) n=0 Div. 16-09. Def f(x) = & an x^² = a + ax+ 9₂x² + 93x²³ +-... is não a power Series Goal: Find the domain Thm: Lef & anx^ be a power Series if X ERH no L Let R. = Then n=1 an ∞ ∞ if x = 0 to if Eanx Conv. abs. Eanx div. If R =∞, then Eanxn Conv. abs. #XER If R=0, then Eanx" Conv. only for X ²0 find the an of f(x) - (the x's for which som Eanx Conv.) I'm suplanta lim n=∞an+1 Let X=1. Then & I n=1 n กร if if Ix > Re R R.O.C. lim 336 XnE+ n=1 Let x=-1. Then Ex^²=(-1)^ <= [¹,1) naln 1x1 <RE R+ and n n+l x "C lim A-B Dive, 7+1 n Harmonic R.O.C Can => 0 n Conv. by AST lim 1=0, an = 1/ zn = antl
Transcribed Image Text:EX. (Boundary points = So (im |an| = lim |+| ns anti EX. Find R.O.C. Notice an = and R=1. check Xx=1: = check x = -1 = f(x) = f(-1) = 28 (-1) ² = 1 - 1 + |-|-|-=-=- C = (-1, 1) n=0 Div. 16-09. Def f(x) = & an x^² = a + ax+ 9₂x² + 93x²³ +-... is não a power Series Goal: Find the domain Thm: Lef & anx^ be a power Series if X ERH no L Let R. = Then n=1 an ∞ ∞ if x = 0 to if Eanx Conv. abs. Eanx div. If R =∞, then Eanxn Conv. abs. #XER If R=0, then Eanx" Conv. only for X ²0 find the an of f(x) - (the x's for which som Eanx Conv.) I'm suplanta lim n=∞an+1 Let X=1. Then & I n=1 n กร if if Ix > Re R R.O.C. lim 336 XnE+ n=1 Let x=-1. Then Ex^²=(-1)^ <= [¹,1) naln 1x1 <RE R+ and n n+l x "C lim A-B Dive, 7+1 n Harmonic R.O.C Can => 0 n Conv. by AST lim 1=0, an = 1/ zn = antl
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