Find the interval of convergence of this power series

[Fernando Rios]

Homework Statement

Find the interval of convergence of each of the following power series; be sure to investigate
the endpoints of the interval in each case.

Relevant Equations

∑((√(x^2+1))^n 2^n/(3^n + n^3))

∑((√(x²+1))ⁿ2²/(3ⁿ+n³))

We use the ratio test:
ρ_n = |2(3ⁿ+n³)√(x²+1)/(3ⁿ⁺¹+(n+1)³)|

ρ = |2√(x²+1)|

ρ < 1

|2√(x²+1)| < 1

No "x" satisfies this expression, so I conclude the series doesn't converge for any "x". However the answer in the book says the series converges for |x| < √(5)/2. What am I dong wrong?

 

[member 587159]

You have to switch numerator and denominator in the ratio test.

Also, what you wrote down is no power series, so the question got the terminology wrong.

 

[Fernando Rios]

You have to switch numerator and denominator in the ratio test.

Also, what you wrote down is no power series, so the question got the terminology wrong.

I wasn't reading the instructions for this problem. Thank you for your answer.