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:

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

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

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

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