Series convergence

Question

∞

Consider the series ∑ 2n/(n+1)!. If the ratio test is applied to this series, which of the following inequalities

n=1

indicates that the series is convergent?

a) lim 2/n+2 < 1

n→ ∞

b) lim 2/(n+2)! < 1

n→ ∞

c) lim n+2/2 < 1

n→ ∞

d) lim 2ⁿ/n+2 < 1

n→ ∞

e) lim 2ⁿ/(n+2)! < 1

n→ ∞

Answer 1

a_n+1 = (2n+2)/(n+2)! a_n = (2n)/(n+1)!

a_n+1 / a_n = (2n+2) / (n+2)! · (n+1)! / (2n) = (n+1) / n(n+2)

However, that ratio does not appear in any of the answers. So my guess is that the series is not 2n/(n+1)!,

but instead 2ⁿ/(n+1)!. If that is the case, then we have a_n+1 / a_n = 2ⁿ⁺¹ / (n+2)! · (n+1)! / (2ⁿ) = 2 / (n+2) and answer choice a is correct.