Determine whether this series is a convergent or divergent.

Question

Determine whether this series is a convergent or divergent.

∑ _n=1^∞ [(n + 1)!]² /(2n + 2)!

Wwould you please explain in steps 😊

Answer 1

∑₁^∞[(n + 1)!]² /(2n + 2)!

∑₁^∞[(n + 1)!]² /2(n + 1)! = ∑₁^∞[(n + 1)!] /2 = ∞

Follow the simplification of factoring 2 out, cancelling (n+1)/(n+1) = 1 and what happens when n goes to infinity.

Answer:

Is Divergent

Answer 2

Fatima:

Let a_{n =} [(n+1)!]²/(2n+2)!

a_{n+1 =} [(n+2)!]²/(2n+4)!

|a_n+1/a_n| = |[(n+2)²(n+1)!²*(2n+2)!]/[(2n+4)*(2n+3)*(2n+2)!*(n+1)!²]

|a_n+1/a_n| = | (n+2)²/[(2n+4)*(2n+3)] |

|a_n+1/a_n| = | (n²+4n+4)/(4n²+14n+12) |

determine n goes to ∞ of |a_n+1/a_n|.

If the limit < 1 we have absolute convergence

if the limit > 1 we have divergence