Question

Determine whether the sequence converges or diverges. If it converges, find the limit

Answer 1

\[a_n= \frac{ (-1)^n * n }{ n^2+6 }\]

Answer 2

I figured the (-1)^n = divergent.... but what happens when it is over infinity?

Answer 3

Well it looks like the numerator alternates back and forth between -n and n. Do you agree?

Answer 4

yes, that is why i said divergent for that one

Answer 5

Good thinking. However, the denominator is of a higher degree, so which will grow quicker, the numerator or the denominator?

Answer 6

\[\lim n--> \inf a_n= \frac{ (-1)^n }{ 1+6/n }\]

Answer 7

im thinking the numerator will

Answer 8

oh wait, thats just one...so denominator will

Answer 9

Well, if the numerator grows with n and the denominator grows with n^2, which one gets bigger faster?

Answer 10

the denominator. but, doesnt the divergent numerator play a role? Or do we ignore that

Answer 11

Well, I'm thinking that, since the denominator grows more quickly than the numerator, the function tends toward zero. The numerator means that the function varies from positive value to negative value to positive value to negative value, etc., all the while getting closer to zero. Make sense?

Answer 12

yes, this is what confuses me so

Answer 13

so, im thinking convergent at zero

Answer 14

or just divergent ... cuz the numerator still throws me off

Answer 15

Me too. Perhaps like this

Answer 16

i see what youre saying, either way, it still gets lower over time

Answer 17

Right on.

Answer 18

sweet, thanks

Answer 19

You're welcome