Determine whether this series is a convergent or divergent. ∑ n=1∞ [(n + 1)!]2 /(2n + 2)!

Question

Determine whether this series is a convergent or divergent.

∑ _n=1^∞ [(n + 1)!]² /(2n + 2)!

PLEASE SOLVE THIS PROBLEM WITH STEPS

Answer 1

Taha:

Use the ratio test.

Find |a_n+1 / a_n |

a_{n =} [(n+1)!]²/(2n+2)!

a_{n+1 =} [(n+2)!]²/(2n+4)!

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

|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) |

I'll let you figure out the limit as n goes to ∞

If the limit < 1 then absolute convergence; if the limit > 1 divergence. See what you get!