## anonymous 2 years ago how do i find the limit as an goes to infinity of the following? (the equation's in the comments)

1. anonymous

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

2. anonymous

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

3. anonymous

we're not sure where to go from here

4. anonymous

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

5. anonymous

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

6. anonymous

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. anonymous

@satellite73 Got this?

9. anonymous

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

10. anonymous

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

11. tkhunny

Whoops. Make that 64, instead of 16.

12. anonymous

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. anonymous

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

14. anonymous

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. anonymous

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

16. anonymous

oh oops duh that makes sense.

17. anonymous

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. anonymous

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

19. anonymous

@satellite73 where did -128 come from?

20. anonymous

sorry probably a dumb question but...

21. anonymous

oh waitn.....nevermind

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