convergence of a sequence

[iDoof]

ok I have this sequence:

((-1^n)*n^3)/(n^3 + 2n^2 + 1)

and I want to know if it converges or diverges. I'm not quite sure how to go about doing this. i know of a theorem that says if you take the limit of the absolute value of the formula, and it equals 0, then the actual formula should also equal 0....but this one's limit of the absolute value equals 1!

i think it diverges though, by instinct, because of the -1^n....so is that it? that it's intuitive like that?


thanks in advance!

 

[tkhunny]

For convergence, your succesive terms have to approach zero.

 

[pka]

“For convergence, your successive terms have to approach zero.”
This is true if this were about series.
However, this question is about sequences.
It does alternate. It has two accumulation points, 1 & −1.
Thus it diverges.

 

[soroban]

Hello, iDoof!

pka is absolutely correct . . . and so are you (it diverges).

. . . . . . . . . . . .n³
. . . (-1)ⁿ -----------------
. . . . . . . .n³ + 2n² + 1


Divide top and bottom by n³:

. . . . . . . . . . . . 1
. . . (-1)ⁿ ------------------
. . . . . . . 1 + 2/n + 1/n³

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
As n -> oo, the fraction goes to: .------------ . = . 1
. . . . . . . . . . . . . . . . . . . . . . . . . . 1 + 0 + 0


But the (-1)ⁿ in front* makes it diverge to +1 <u>or</u> -1.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

* . I call it the "blinker" . . . which has many uses.

The sequence: .5, 7, 5, 7, 5, 7, . . . .can be expressed: .a_n .= .6 + (-1)ⁿ


If a sequence alternates between two values (or functions) A and B:

. . . . . . . . .1 - (-1)ⁿ . . . . 1 + (-1)ⁿ
. . . f_n . = . ---------- A .+ .----------- B
. . . . . . . . . . . 2 . . . . . . . . . .2

 

[tkhunny]

This is true if this were about series.

Did I do that AGAIN? Grrr...