Question
Asked Dec 3, 2019
If k is a positive integer, find the radius of convergence, R, of the series
Σ
(n!)k+2
-xn
((k2)n)!
n 0
R =
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If k is a positive integer, find the radius of convergence, R, of the series Σ (n!)k+2 -xn ((k2)n)! n 0 R =

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Expert Answer

Step 1

It is required to calculate the radius of convergence R of the series:

k+2
(n!)*
(k + 2)n)!"
n=0
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k+2 (n!)* (k + 2)n)!" n=0

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Step 2

Here,

(n!)*2
((k+2)n)
((n+1))
(k+2)(n+1)!
k+2
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(n!)*2 ((k+2)n) ((n+1)) (k+2)(n+1)! k+2

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Step 3

Now, find the ratio for...

(n+1)!)**
((k+2)(n+1)
k+2
((n+1)*
((k+2)(n+1))
!
(k+2)n)!
(n)o
n+1
k+2
(п!)
а,
k+2
((k+2)n)!
(n+1)(n)
(n+1)!
k+2
x
k+2
(nt)3
(n+1)*x
(n+1)!
k+2
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(n+1)!)** ((k+2)(n+1) k+2 ((n+1)* ((k+2)(n+1)) ! (k+2)n)! (n)o n+1 k+2 (п!) а, k+2 ((k+2)n)! (n+1)(n) (n+1)! k+2 x k+2 (nt)3 (n+1)*x (n+1)! k+2

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