What is the interval of convergence of #sum(2^n(x-3)^n)/(sqrt(n+3))#?

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Answer 1 · 2017-04-05T14:14:22

#x in (5/2, 7/2)#

Ratio test:

#lim_(n to 0) abs( ((2^(n+1)(x-3)^(n+1))/(sqrt(n+4)))/((2^n(x-3)^n)/(sqrt(n+3))))#

#=lim_(n to 0) abs( (2(x-3)sqrt(n+3))/(sqrt(n+4)))#

#=lim_(n to 0) abs( (2(x-3)sqrt(1+3/n))/(sqrt(1+4/n)))#

#= 2 abs ((x-3)) lt 1 #

Which means either:

#- 2 (x-3) lt 1 implies x gt 5/2#

# 2 (x-3) lt 1 implies x lt 7/2#

#implies x in (5/2, 7/2)#