Question

how do i find the limit as an goes to infinity of the following? (the equation's in the comments)

Answer 1

\[\frac{ 4 }{ n }\sum_{a=1}^{n}(-(\frac{ 4a }{ n })^{2} -4(\frac{ 4a }{ n }))\] This is the equation in question

Answer 2

\[\frac{ 4 }{ n }\sum_{a=1}^{n}(-(\frac{ 4a }{ n })^{2}) - \frac{ 4 }{ n }\sum_{a=1}^{n}(4(\frac{ 4a }{ n}))\]

Answer 3

we're not sure where to go from here

Answer 4

looks like you are trying to compute some sort or reimann sum

Answer 5

exactly. a right riemann sum of -x^2 -4x to be exact.

Answer 6

don't let the \(n\) through you off in the summation
you are summing over \(a\) (never seen an \(a\) used like that before, but no matter)
pull it right out front of the sum
then use the formula for
\[\sum_{a=1}^na^2\] and also \[\sum_{a=1}^na\]

Answer 7

After a little algebra.

\(-\dfrac{16}{n^{2}}\left[\dfrac{1}{n}\sum\limits_{a=1}^{n}a^{2} + \sum\limits_{a=1}^{n}a\right]\)

Answer 8

@satellite73 Got this?

Answer 9

\[\frac{ 4 }{ n }\sum_{a=1}^{n}(-(\frac{ 4a }{ n })^{2})\]
\[=-\frac{64}{n^3}\sum_{a=1}^na^2\] for example
@primeralph maybe, i am tired and all this latex is wearing me out

Answer 10

@tkhunny i think the \(4\) is being squared as well

Answer 11

Whoops. Make that 64, instead of 16.

Answer 12

not really sure the sum is correct, but assuming it is for the moment
\[-\frac{64}{n^3}\sum_{k=1}^nk^2=-\frac{64}{n^3}\frac{n(n+1)(2n+1)}{6}\] for the first one

Answer 13

Wait but how does this help us get the limit as n -> infinity?

Answer 14

taking the limit as \(n\to \infty\) is very easy because the numerator is a polynomial of degree 3 will leading coefficient -128 and the denominator is a polynomial of degree 3 with leading coefficent 6

Answer 15

as you remember from some pre calc course about "horizontal asymptotes" that makes the limit \(-\frac{128}{6}\)

Answer 16

oh oops duh that makes sense.

Answer 17

easy if you think about it that way instead of the way they do it in the text by dividing top and bottom by \(n^3\) and all that other nonsense

Answer 18

Thanks so much!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Answer 19

@satellite73 where did -128 come from?

Answer 20

sorry probably a dumb question but...

Answer 21

oh waitn.....nevermind