Question

Does this series converge or diverge?
(n*2^n)/(2^n + 3^n) and by which test

Answer 1

you can simply take values or take the limit as x --> infinity^_^

Answer 2

the limit as x-->infinite is zero, which is inconclusive

Answer 3

no no, it means that the series converge ^_^

Answer 4

when you end up with zero, it means the series is converging :) and when you end up with infinity it diverges

Answer 5

the series will not necessarily converge when the limit of the sequence as n-->infinite is zero, for example 1/n

Answer 6

Sorry sstarica, that is not correct

Answer 7

I need another test

Answer 8

prifk is right

Answer 9

i was thinking limit comparison or ratio

Answer 10

but ratio yields "1", inconclusive.. and not sure what converging or diverging series to limit compare to

Answer 11

Simplifying a bit will help here..\[(n*2^n)/(2^n + 3^n) = n/[1 + (3/2)^n]\]

Answer 12

yes, I did that. not sure what the next steps are ... l'hopitals rule says the limit goes to zero

Answer 13

perhaps a comparison of some sort?

Answer 14

this question was driving me mad and I'm a tutor lol!

Answer 15

You tried the ratio test?

Answer 16

yes. 1

Answer 17

i'll try again

Answer 18

yeah, still 1

Answer 19

converge.
Ask hinted by polpak, write (n*2^n)/(2^n + 3^n) as n/[1 + (3/2)^n].
Then compare this with n(2/3)^n which is convergent by the Ratio test

Answer 20

Nice!

Answer 21

well done! thank you