Quantcast

A community for students. Sign up today!

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

aef321

  • one year ago

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

  • This Question is Closed
  1. alffer1
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  2. alffer1
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  3. alffer1
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    we're not sure where to go from here

  4. satellite73
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

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

  5. alffer1
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  6. satellite73
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    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
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    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
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @satellite73 Got this?

  9. satellite73
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

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

  10. satellite73
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

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

  11. tkhunny
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Whoops. Make that 64, instead of 16.

  12. satellite73
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    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
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  14. satellite73
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    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
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

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

  16. alffer1
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    oh oops duh that makes sense.

  17. satellite73
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    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
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  19. alffer1
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @satellite73 where did -128 come from?

  20. alffer1
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    sorry probably a dumb question but...

  21. alffer1
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    oh waitn.....nevermind

  22. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Ask a Question
Find more explanations on OpenStudy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.