Hello, rcmango!

Converges or diverges?

SUM n_infinity (-1)^n(2^n*n!)/(5*8*11*...*(3n +2)

I would use the Ratio Test

an+1 . . . . .2^n+1·(n+1)! . . .5·8·11···(3n+2) . . . .2(n + 1) . . . . 2n + 2

------ . = . ------------------ · ------------------ . = . ---------- . = . ---------

. an . . . . .5·8·11···(3n+5) . . . 2^n·n! . . . . . . . . 3n + 5 . . . . . 3n + 5

. . . . . . . . . . . . . . . . . . . . . 2 + 2/n

Divide top and bottom by n: . ---------

. . . . . . . . . . . . . . . . . . . . . 3 + 5/n

Now take the limit as n → ∞