by cancelling the 2n^2 and n^2 did i do it wrong?

as you said they have to be product to cancel out?

......n² + 2n + 2 . . .2n² + 1
. . .--------------- . .. ----------
. . . . . . . . . . . . . .

=
...n²/ + 2n/ + 2/ . ... .2n²/ + 1/
. . .--------------------------- . .. ------------------
. . . . ....... . n²/
. . . . . .......... . .n²/

=
...1 + 2/n + 2/ . ............... .2 + 1/
. . .--------------------------- . .. ------------------
. . . . ....... . 1
. . . . . .......... ........ .1
=
(1 + 2/n + 2/n²)(2 + 1/)

take the limit as n-->oo we get

= (1 + 0 + 0)(2 + 0)
= 2

2. sorry I forgot to quote what I was asking

here the question:

(Un + 1) / (Un) = 1 + 2n^2 + 4n + 2 / 1 + n^2 + 2n + 1 . 1 + n^2 / 1 + 2n^2

(Un + 1) / (Un) = 2n^2 + 4n + 3 / n^2 + 2n + 2 . 1 + n^2 / 1 + 2n^2

cancelling out now to give: by cancelling the 2n^2 and n^2 did i do it wrong?

(Un + 1) / (Un) = 4n + 3 / 2n + 2

Lim n → ∞
(Un + 1) / (Un) = 4 + (3/n) / 2 + (2/n)

= 4 + 0 / 2 + 0

= 2

as > 1 series diverges.

that's how I did it..

sorry I forgot to quote what I was asking

here the question:

(Un + 1) / (Un) = 1 + 2n^2 + 4n + 2 / 1 + n^2 + 2n + 1 . 1 + n^2 / 1 + 2n^2

(Un + 1) / (Un) = 2n^2 + 4n + 3 / n^2 + 2n + 2 . 1 + n^2 / 1 + 2n^2

cancelling out now to give: by cancelling the 2n^2 and n^2 did i do it wrong?

(Un + 1) / (Un) = 4n + 3 / 2n + 2

Lim n → ∞
(Un + 1) / (Un) = 4 + (3/n) / 2 + (2/n)

= 4 + 0 / 2 + 0

= 2

as > 1 series diverges.

that's how I did it..
yes, you're correct. you might want to work on your notation though. but i'm guilty of that as well

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