Determine the disk of convergence D(20, R) of the following power series: (n!)2 2"; (2n)! (z + 3)" (n + 1)2" (a) (b) n=0 (c) (d) E(-1)"2n 22n 2" ; n n=0
Determine the disk of convergence D(20, R) of the following power series: (n!)2 2"; (2n)! (z + 3)" (n + 1)2" (a) (b) n=0 (c) (d) E(-1)"2n 22n 2" ; n n=0
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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![The power series in C with center zo and coefficients an E C is the series
Σ
An (z – z0)" ,
(1)
n=0
where z is taken in some domain containing 20. In class, we have seen that there exists
RE [0, 00] such that (1) converges absolutely at all z E D(zo, R) (open disk centered at zo
with radius R) and uniformly-absolutely in D(2o,r) for all r < R, while it does not converge
for |z – zo| > R. Such R is called the radius of convergence of (1), and D(20, R) is the
disk of convergence.
Determine the disk of convergence D(20, R) of the following power series:
(n!)?
(а)
(b) (2+3)"
Σ
(2n)!";
(n + 1)2"
n=0
n=0
22n
( d) Σ-1)"5-2n
(c)
n
2n
n=
n=0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15ed467-90ec-4e60-afef-3d3f6119f74d%2Fd5e4b2e9-516e-4020-9dea-127060ac5fb5%2Fnqw0hl_processed.png&w=3840&q=75)
Transcribed Image Text:The power series in C with center zo and coefficients an E C is the series
Σ
An (z – z0)" ,
(1)
n=0
where z is taken in some domain containing 20. In class, we have seen that there exists
RE [0, 00] such that (1) converges absolutely at all z E D(zo, R) (open disk centered at zo
with radius R) and uniformly-absolutely in D(2o,r) for all r < R, while it does not converge
for |z – zo| > R. Such R is called the radius of convergence of (1), and D(20, R) is the
disk of convergence.
Determine the disk of convergence D(20, R) of the following power series:
(n!)?
(а)
(b) (2+3)"
Σ
(2n)!";
(n + 1)2"
n=0
n=0
22n
( d) Σ-1)"5-2n
(c)
n
2n
n=
n=0
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