Let S(x)=\sum_{n=1}^\infty\frac{4^n(x+4)^{2n}}{n} Find the radius of convergence.

Ayden Case 2022-02-22 Answered
Let
S(x)=n=14n(x+4)2nn
Find the radius of convergence.
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Expert Answer

Cicolinif73
Answered 2022-02-23 Author has 7 answers
Let SN(x) denote the partial sums of the series S(x)=n=14n(x+2)2nn. Then, letting y=(2x+8)2, with |y|<1, we have
SN(x)=n=1N4n(x+2)2nn
=n=1Nynn
=n=1N0yzn1dz
=0y1zN1zdz
log|1y| as N (1)
=log|1(2x+8)2|
=log|2x+9|log|2x+7|
where the justification for interchanging the limit with the integral in (1) is provided by the Dominated Convergence Theorems since |1zN1z|2|1z| for |z|<1
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