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Suppose you know that the series Σobn" converges for |x| < 2. =0 (a) Draw the largest open interval of convergence of the series (b) Is the series n=0 bn n+1 convergent? (c) How about the series nbn? Is it convergent? n=1 8 n=0 bn n+1 √x+1 on a number line.

Suppose you know that the series Σobn" converges for |x| < 2. =0 (a) Draw the largest open interval of convergence of the series (b) Is the series n=0 bn n+1 convergent? (c) How about the series nbn? Is it convergent? n=1 8 n=0 bn n+1 √x+1 on a number line.

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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Need help with this question a b c

ao+a₁(x−c)+α₂(x-c)²+ has radius of convergence - R where R > 0 or R = ∞, then the function f(x) = Σax(x − c)* = ao+a₁(x− c) + a₂(x − c)² +….. k=0 is differentiable on (c- R,c + R). On that interval, Theorem. If the power series Σak(x−c)* = = ∞ (a) f'(x) = Σ kak(x − c)k−¹ = a₁ + 2a2(x − c) + 3a3(x − c)² + ……. k=1 a ) dx = C + Σ −(x − c)k+¹ = C + ao(x − c) + 2/(x- c)² + 3 k=0 (b) f(x) dx k=0 The power series in (1) and (2) both have radius of convergence R. (Note: Although the radius of convergence remains unchanged, the interval of convergence may change.) Suppose you know that the series obna converges for x < 2. (b) Is the series ak k+1 (a) Draw the largest open interval of convergence of the series n=0 bn n+1 convergent? ∞ (c) How about the series nbn? Is it convergent? n=1 ∞ n=0 bn n+1 -(x − c)³ +. -x+1 number line.
Transcribed Image Text:ao+a₁(x−c)+α₂(x-c)²+ has radius of convergence - R where R > 0 or R = ∞, then the function f(x) = Σax(x − c)* = ao+a₁(x− c) + a₂(x − c)² +….. k=0 is differentiable on (c- R,c + R). On that interval, Theorem. If the power series Σak(x−c)* = = ∞ (a) f'(x) = Σ kak(x − c)k−¹ = a₁ + 2a2(x − c) + 3a3(x − c)² + ……. k=1 a ) dx = C + Σ −(x − c)k+¹ = C + ao(x − c) + 2/(x- c)² + 3 k=0 (b) f(x) dx k=0 The power series in (1) and (2) both have radius of convergence R. (Note: Although the radius of convergence remains unchanged, the interval of convergence may change.) Suppose you know that the series obna converges for x < 2. (b) Is the series ak k+1 (a) Draw the largest open interval of convergence of the series n=0 bn n+1 convergent? ∞ (c) How about the series nbn? Is it convergent? n=1 ∞ n=0 bn n+1 -(x − c)³ +. -x+1 number line.
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