## aef321 one year ago how do i find the limit as an goes to infinity of the following? (the equation's in the comments)

1. alffer1

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

2. alffer1

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

3. alffer1

we're not sure where to go from here

4. satellite73

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

5. alffer1

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

6. satellite73

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 $\sum_{a=1}^na$

7. tkhunny

After a little algebra. $$-\dfrac{16}{n^{2}}\left[\dfrac{1}{n}\sum\limits_{a=1}^{n}a^{2} + \sum\limits_{a=1}^{n}a\right]$$

8. primeralph

@satellite73 Got this?

9. satellite73

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

10. satellite73

@tkhunny i think the $$4$$ is being squared as well

11. tkhunny

Whoops. Make that 64, instead of 16.

12. satellite73

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

13. alffer1

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

14. satellite73

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

15. satellite73

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

16. alffer1

oh oops duh that makes sense.

17. satellite73

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

18. aef321

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

19. alffer1

@satellite73 where did -128 come from?

20. alffer1

sorry probably a dumb question but...

21. alffer1

oh waitn.....nevermind