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

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

Mathematics
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you can simply take values or take the limit as x --> infinity^_^
the limit as x-->infinite is zero, which is inconclusive
no no, it means that the series converge ^_^

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Other answers:

when you end up with zero, it means the series is converging :) and when you end up with infinity it diverges
the series will not necessarily converge when the limit of the sequence as n-->infinite is zero, for example 1/n
Sorry sstarica, that is not correct
I need another test
prifk is right
i was thinking limit comparison or ratio
but ratio yields "1", inconclusive.. and not sure what converging or diverging series to limit compare to
Simplifying a bit will help here..\[(n*2^n)/(2^n + 3^n) = n/[1 + (3/2)^n]\]
yes, I did that. not sure what the next steps are ... l'hopitals rule says the limit goes to zero
perhaps a comparison of some sort?
this question was driving me mad and I'm a tutor lol!
You tried the ratio test?
yes. 1
i'll try again
yeah, still 1
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
Nice!
well done! thank you

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