Converges or diverges?

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

i believe this is an alternating series.

the 5*8*11*.. confuses me.

thankyou for any help.

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- Apr 18th 2007, 12:02 AMrcmangoalternate series question
Converges or diverges?

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

i believe this is an alternating series.

the 5*8*11*.. confuses me.

thankyou for any help. - Apr 18th 2007, 02:53 AMSoroban
Hello, rcmango!

Quote:

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 → ∞

- Apr 18th 2007, 10:37 AMCaptainBlack
Now this is overkill, we have proven that the series is absolutely convergent.

We could have proven that it converges (without proving that it is

absolutely convergent) by using the Alternating Series test.

This requires that we show that the absolute value of the terms is

eventually decreasing, and the limit of the terms is 0.

RonL - Apr 18th 2007, 10:39 AMCaptainBlack